On a Class of Nondifferentiable Multiobjective Continuous Programming Problems

Abstract

Fritz John and Karush-Kuhn-Tucker type optimality conditions for a nondifferentiable multiobjective variational problem with equality and inequality constraints are obtained. Using Karush-Kuhn-Tucker type optimality conditions, a Wolfe type second-order nondifferentiable multiobjective dual variational problem is constructed. Various duality results for the pair of Wolfe type second-order dual variational problems are proved under second-order pseudoinvexity. A pair of Wolfe type dual variational problems having equality and inequality constraints with natural boundary values is also formulated to prove various duality results. Finally, the linkage between our results and those of their static counterparts existing in the literature is briefly outlined.

1 Introduction

Second-order duality in mathematical programming has been widely researched. As second-order dual to a constrained optimization problem gives a tighter bound and hence enjoys computational advantage over the first-order dual to the problem. Mangasarian [1] was the first to second-order duality in non-linear programming. Motivated with analysis of Mangasarian [1], Chen [2] presented Wolfe type second-order dual to a class of constrained variational problems under an involved invexity like conditions. Later Husain et al. [3] introduced second-order invexity and generalized invexity and presented a Mond-Weir type second-order dual to the problem of [2] in order to relax implicit invexity requirements to the generalized second order invexity.

Multiobjective optimization has applications in various fields of science that includes engineering, economics and logistics where optimal decisions need to be taken in the presence of trade-off between two or more conflicting objectives. Motivated with facts that multiobjective optimization models represent a variety of real life problems, Husain and Jain [4] formulated a multiobjective version of the variational problem considered by Chen [2] with equality and inequality constraints which represent more realistic problems than those variational problems with an inequality constraint only. In [4], they studied optimality and duality for their variational problem. In this paper, optimality conditions for a multiobjective variational problem which contains a term of square root of certain quadratic form in each component of the vector valued objective function of the problem, are obtained and Wolfe type second-order duality is investigated for this problem. Finally, our results are shown to be the dynamic generalization of those of nondifferentiable multiobjective mathematical programming problems, already studied in the literature.

2 Definitions and Related Pre-requisites

Let I = [a, b] be a real interval, $\varLambda$ and $\phi :\,I \times R^{n} \times R^{n} \to R^{m}$ be twice continuously differentiable functions. In order to consider $\phi \left( {t,x\left( t \right),\dot{x}\left( t \right)} \right),$ where $x :\,I \to R^{n}$ is differentiable with derivative $\dot{x} ,$ denoted by $\phi_{x}$ and $\phi_{{\mathop x\limits^{.}}} ,$ the first order derivatives of ϕ with respect to x(t) and $\dot{x}\left( t \right) ,$ respectively, that is,

$$ \phi_{x} = \left( {\frac{\partial \phi }{{\partial x^{1} }},\frac{\partial \phi }{{\partial x^{2} }}, \cdots ,\frac{\partial \phi }{{\partial x^{n} }}} \right)^{T} ,\;\phi_{{\mathop x\limits^{.}}} = \left( {\frac{\partial \phi }{{\partial \dot{x}^{1} }},\frac{\partial \phi }{{\partial \dot{x}^{2} }}, \cdots ,\frac{\partial \phi }{{\partial \dot{x}^{n} }}} \right)^{T} $$

Further Denote by ϕ _xx and ψ _x the n × n Hessian matrix and m × n Jacobian matrix respectively. The symbols $\phi_{{\mathop x\limits^{.}}} ,\phi_{{\dot{x}x}} ,\phi_{{x\dot{x}}}$ and $\psi_{{\mathop x\limits^{.}}}$ have analogous representations.

Designate by X, the space of piecewise smooth functions $x :I \to R^{n} ,$ with the norm $\left\| x \right\| = \left\| x \right\|_{\infty} + \left\| {Dx} \right\|_{\infty },$ where the differentiation operator D is given by

$$ u = Dx \Leftrightarrow x\left( t \right) = \int\limits_{a}^{t} {u\left( s \right)} ds, $$

Thus $\frac{d}{dt} = D$ except at discontinuities.

We incorporate the following definitions which are required for the derivation of the duality results.

Definition 1 (Second - order Invex):

If there exists a vector function $\eta = \eta \left( {t,x,\bar{x}} \right) \in R$ where $\eta :I \times R^{n} \times R^{n} \to R^{n}$ and with η = 0 at t = a and t = b such that for a scalar function $\phi \left( {t,x,\dot{x}} \right) ,$ the functional $\int\limits_{I} {\phi \left( {t,x,\dot{x}} \right)} dt$ where $\phi :\,I \times R^{n} \times R^{n} \to R$ satisfies

$$ \begin{aligned} & \int\limits_{I} \phi \left( {t,x,\dot{x}} \right)dt - \int\limits_{I} {\left\{ {\phi \left( {t,\bar{x},\dot{\bar{x}}} \right) - \frac{1}{2}\beta \left( t \right)^{T} G\beta \left( t \right)} \right\}dt} \\ & \quad \ge \int\limits_{I} {\left\{ {\eta ^{T} \phi _{x} \left( {t,\bar{x},\dot{\bar{x}}} \right) + \left( {D\eta } \right)^{T} \phi _{{\mathop x\limits^{.}}} \left( {t,\bar{x},\dot{\bar{x}}} \right) + \eta ^{T} G\beta \left( t \right)} \right\}dt} \\ \end{aligned} $$

then $\int\limits_{I} {\phi \left( {t,x,\dot{x}} \right)} dt$ is second-order invex with respect to η,where $G = \phi_{xx} - 2D\phi_{{x\dot{x}}} + D^{2} \phi_{{\dot{x}\dot{x}}} - D^{3} \phi_{{\dot{x}\ddot{\textit{x}}}}$ and $\beta \in C\left( {I,R^{n} } \right)$ the space of n-dimensional continuous vector functions.

Definition 2 (Second - order Pseudoinvex):

If the functional $\int\limits_{I} {\phi \left( {t,x,\dot{x}} \right)} dt$ satisfies

$$\begin{aligned}& \int\limits_{I} {\left\{ {\eta^{T} \phi_{x} + \left( {D\eta } \right)^{T} \phi_{{\mathop x\limits^{.}}} + \eta^{T} G\beta \left( t \right)} \right\}} dt \ge 0 \\ & \quad \Rightarrow \int\limits_{I} \phi \left( {t,x,\dot{x}} \right)dt \ge \int\limits_{I} {\left\{ {\phi \left( {t,\bar{x},\dot{\bar{x}}} \right) - \frac{1}{2}\beta^{T} \left( t \right)G\beta \left( t \right)} \right\}dt}\\ \end{aligned} $$

then $\int\limits_{I} {\phi \left( {t,x,\dot{x}} \right)} dt$ is said to be second-order pseudoinvex with respect to η.

Definition 3 (Second - order strict - pseudoinvex):

If the functional $\int\limits_{I} {\phi \left( {t,x,\dot{x}} \right)\,} dt$ satisfies

$$\begin{aligned} & \int\limits_{I} {\left\{ {\eta^{T} \phi_{x} + \left( {D\eta } \right)^{T} \phi_{{\mathop x\limits^{.}}} + \eta^{T} G\beta \left( t \right)} \right\}} dt \ge 0 \\ & \quad \Rightarrow \int\limits_{I} \phi \left( {t,x,\dot{x}} \right)dt > \int\limits_{I} {\left\{ {\phi \left( {t,\bar{x},\dot{\bar{x}}} \right) - \frac{1}{2}\beta \left( t \right)^{T} G\beta \left( t \right)} \right\}dt}\\ \end{aligned} $$

then $\int\limits_{I} {\phi \left( {t,x,\dot{x}} \right)} dt$ is said to be second-order pseudoinvex with respect to η.

The following inequality will also be required in the forthcoming analysis of the research:

It states that

$$ x\left( t \right)^{T} B\left( t \right)z\left( t \right) \le \left( {x\left( t \right)^{T} B\left( t \right)x\left( t \right)} \right)^{1/2} \left( {z\left( t \right)^{T} B\left( t \right)z\left( t \right)} \right)^{1/2} $$

with equality in above if $B\left( t \right)x\left( t \right) - q\left( t \right)z\left( t \right) = 0,$ for some $q\left( t \right) \in R,t \in I.$

Throughout the analysis of this research, the following conventions for the inequalities will be used:

If $d,s \in R^{n}$ with $d = \left( {d^{1} ,d^{2} , \ldots ,d^{n} } \right)$ and $s = \left( {s^{1} ,s^{2} , \ldots ,s^{n} } \right),$ then

$$ \begin{aligned} d & { \geqq }s \Leftrightarrow d^{i} { \geqq }s^{i} ,\quad \left( {i = 1,2, \ldots ,n} \right) \\ d & \ge s \Leftrightarrow d{ \geqq }s\quad and\quad d \ne s \\ d & > s \Leftrightarrow d^{i} > s^{i} ,\quad \left( {i = 1,2, \ldots ,n} \right) \\ \end{aligned} $$

3 Optimality Conditions

We consider the following nondifferentiable multiobjective variational problems containing terms of square root functions.

(NWVEP): Minimize

$$\begin{aligned} & \left( {\int\limits_{I} {\left\{ {f^{1} \left( {t,x\left( t \right),\dot{x}\left( t \right)} \right) + \left( {x\left( t \right)^{T} B^{1} \left( t \right)x\left( t \right)} \right)^{{1/2}} } \right\}} dt, \ldots ,} \right. \\ & \left. { \quad \int\limits_{I} {\left\{ {f^{p} \left( {t,x\left( t \right),\dot{x}\left( t \right)} \right) + \left( {x\left( t \right)^{T} B^{p} \left( t \right)x\left( t \right)} \right)^{{1/2}} } \right\}} dt} \right) \\ \end{aligned} $$

subject to

$$ x\left( a \right) = 0, \quad x\left( b \right) = 0 $$

(1)

$$ g\left( {t,x\left( t \right),\dot{x}\left( t \right)} \right) \leqq 0, \quad t \in I $$

(2)

$$ h\left( {t,x\left( t \right),\dot{x}\left( t \right)} \right) = 0, \quad t \in I $$

(3)

where

• (i) $f^{i} :I \times R^{n} \times R^{n} \to R,i \in K = \left\{ {1,2, \ldots ,p} \right\},g :\,I \times R^{n} \times R^{n} \to R^{m}$ and $h :\,I \times R^{n} \times R^{n} \to R^{k}$ are assumed to be continuously differentiable functions, and

• (ii) for each $t \in I,i \in K = \left\{ {1,2, \ldots ,p} \right\},B^{i} \left( t \right)$ is an n × n positive semi definite (symmetric) matrix, with $B^{i} \left( . \right)$ continuous on i.

If B ⁱ (t) = 0 for all i and t, then the above problem reduces to the problem of [4].

In order to obtain optimality condition for the problem (NWVEP) we consider the following nondifferentiable single objective continuous programming problem considered by Chandra et al. [5]:

$$\begin{array}{*{20}l} {{\text{(}}{\mathbf{CP}}{\text{):}}\quad\, {\text{Minimize}}} \hfill & {\int\limits_{I} \phi \left( {t,x\left( t \right),\dot{x}\left( t \right)} \right)dt} \hfill \\ {\quad \quad \quad \; {\text{Subject}}\,{\text{to}}} \hfill & {x\left( a \right) = 0,\quad x\left( b \right) = 0} \hfill \\ {} \hfill & {g\left( {t,x\left( t \right),\dot{x}\left( t \right)} \right){ \leqq }0,\;h\left( {t,x\left( t \right),\dot{x}\left( t \right)} \right) = 0,\quad t \in I} \hfill \\ \end{array} $$

where $\phi :\,I \times R^{n} \times R^{n} \to R$ is a continuous differentiable functions, and g and h as the same as given for (NWVEP).

Chandra et al. [5] derive the following Fritz John type optimality conditions:

Lemma 1 (Fritz John type optimality conditions):

If $\bar{x}$ is an optimality solution of the problem (NWVEP), and if $h_{x} \left( {.,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right)$ maps X onto a closed subspace of $C\left( {I,R^{k} } \right) ,$ then there exist Lagrange multipliers $\tau \in R,$ piecewise smooth $\bar{y} :\,I \to R^{m}$ and $\bar{z} :\,I \to R^{k}$ such that

$$ \begin{aligned} & \tau f_{x} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right) + \bar{y}\left( t \right)^{T} g_{x} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right) + \bar{z}\left( t \right)^{T} h_{x} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right) \\ & \quad - D\left( {\tau f_{{\mathop x\limits^{.}}} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right) + \bar{y}\left( t \right)^{T} g_{{\mathop x\limits^{.}}} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right)} \right. \\ & \left. {\quad + \bar{z}\left( t \right)^{T} h_{{\mathop x\limits^{.}}} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right)} \right) = 0,\quad t \in I \\ & \bar{x}\left( t \right)^{T} B\left( t \right)w\left( t \right) = \left( {\bar{x}\left( t \right)^{T} B\left( t \right)\bar{x}\left( t \right)} \right)^{{1/2}} ,\quad t \in I \\ & w\left( t \right)^{T} B\left( t \right)w\left( t \right){ \leqq }1,\quad t \in I \\ & y\left( t \right)^{T} g\left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right) = 0,\quad t \in I \\ & \left( {\tau ,y\left( t \right)} \right){ \geqq }0,\left( {\tau ,y\left( t \right),z\left( t \right)} \right) \ne 0,\quad t \in I \\ \end{aligned} $$

The above Fritz-John type necessary optimality conditions if $\tau = 1$ (then $\bar{x}$ may called normal). For $\tau = 1 ,$ it suffice to assume that the Robinson conditions [5] holds or Slater’s condition [5] holds, i.e. for some $v \in X$ and all $t \in I ,$

$$ t \in Ig\left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right) + g_{x} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right)v\left( t \right) + g_{{\mathop x\limits^{.}}} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right)\dot{v}\left( t \right) > 0. $$

The following lemma relates an efficient of (NWVEP) with an optimal solution of p-single objective variational problems.

Lemma 2 (Chankong and Haimes):

A point $\bar{x}\left( t \right) \in X$ is an efficient solution of (NWVEP) if and only $\bar{x}\left( t \right)$ is an optimal solution (NWVEP_r) for each $r \in K = \left\{ {1,2, \ldots ,p} \right\}$

$\begin{aligned} {\text{(NWVEP}}_{{\text{r}}} {\text{):}}\quad& {\text{Minimize}}\quad \int\limits_{I} {\left( {f^{r} \left({t,x(t),\dot{x}(t)} \right) + \left( {x(t)^{T} B^{r} (t)x(t)}\right)^{{1/2}} } \right)} dt \\ & {\text{subject}}\;{\text{to}}\\ & \quad \quad \quad \quad x\left( a \right) = 0,\quad x\left(b \right) = 0 \\ & \quad \quad \quad \quad g\left( {t,x\left( t\right),\dot{x}\left( t \right)} \right)\leqq 0,h\left( {t,x\left( t\right),\dot{x}\left( t \right)} \right) = 0,\quad t \in I \\ &\quad \quad \quad \quad \int\limits_{I} {\left( {f^{i} \left({t,x\left( t \right),\dot{x}\left( t \right)} \right) + \left({x\left( t \right)^{T} B^{i} \left( t \right)x\left( t \right)}\right)^{{1/2}} } \right)dt} \\ & \quad \quad \quad \quad\,\leqq \int\limits_{I} {\left( {f^{i} \left( {t,\bar{x}\left( t\right),\dot{\bar{x}}\left( t \right)} \right) + \left({\bar{x}\left( t \right)^{T} B^{i} \left( t \right)\bar{x}\left( t\right)} \right)^{{1/2}} } \right)dt} , \\ & \quad \quad \quad\quad \quad i \in K_{r} = K - \left\{ r \right\} \\ \end{aligned}$

Since the variational problem (NWVEP) does not contain integral inequality, it can easily be shown that Lemma 2 still remains valid for the constraint without integral sign. That is,

$$ f^{i} \left( {t,x\left( t \right),\dot{x}\left( t \right)} \right) + \left( {x\left( t \right)^{T} B^{i} \left( t \right)x\left( t \right)} \right)^{1/2} { \leqq }f^{i} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right) + \left( {\bar{x}\left( t \right)^{T} B^{i} \left( t \right)\bar{x}\left( t \right)} \right)^{1/2} , \quad i \in K_{r} $$

Theorem 1 (Fritz John type optimality conditions):

Let $\bar{x}\left( t \right)$ be an efficient solution of (NWVEP) and if $h_{x} \left( {.,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right)$ map X onto a closed subspace of $C\left( {I,R^{k} } \right).$ Then there exist $\bar{\lambda }^{i} \in R$ and piecewise smooth functions $\bar{y} :\,I \to R^{m} ,$ $\bar{z} :\,I \to R^{k}$ and $w^{i} :\,I \to R^{m} ,i \in K$ such that

$$\begin{aligned} & \sum\limits_{i = 1}^{p} {\lambda^{i} \left( {f_{x}^{i} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right) + B^{i} \left( t \right)w^{i} \left( t \right) - Df_{{\mathop x\limits^{.}}}^{i} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right)} \right)} \\& \quad + y\left( t \right)^{T} g_{x} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right) + z\left( t \right)^{T} h_{x} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right) \\ &\quad - D\left( {y\left( t \right)^{T} g_{{\mathop x\limits^{.}}} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right) + z\left( t \right)^{T} h_{{\mathop x\limits^{.}}} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right)} \right) = 0,\quad t \in I \\ & \bar{x}\left( t \right)^{T} B^{i} \left( t \right)w^{i} \left( t \right) = \left( {\bar{x}\left( t \right)^{T} B^{i} \left( t \right)\bar{x}\left( t \right)} \right)^{1/2} ,\left( {w^{i} \left( t \right)^{T} B^{i} \left( t \right)w^{i} \left( t \right)} \right)^{1/2} { \leqq }1,\quad i \in K,t \in I \\ & \left( {\bar{\lambda }^{1} ,\bar{\lambda }^{2} , \ldots ,\bar{\lambda }^{p} ,y\left( t \right)} \right) \geqq 0,\left( {\bar{\lambda }^{1} ,\bar{\lambda }^{2} , \ldots ,\bar{\lambda }^{p} ,y\left( t \right),z\left( t \right)} \right) \ne 0,\quad t \in I \\ \end{aligned}$$

Proof:

Since $\bar{x}\left( t \right)$ is an efficient solution of (NWVEP), by Lemma 1, $\bar{x}\left( t \right)$ is an optimal solution of (NWVEP_r) for each $r \in K$ and hence of (NWVEP₁). Hence by Lemma 2, there exist $\bar{\lambda }^{i} \in R,\;i \in K$ and piecewise smooth functions $\bar{y} :\,I \to R^{m} ,$ $\bar{z} :\,I \to R^{k}$ such that

$$\begin{aligned} & \sum\limits_{i = 1}^{p} {\lambda^{i} \left( {f_{x}^{i} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right) + B^{i} \left( t \right)w^{i} \left( t \right) - Df_{{\mathop x\limits^{.}}}^{i} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right)} \right)} \\ &\quad + y\left( t \right)^{T} g_{x} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right) + z\left( t \right)^{T} h_{x} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right) \\ &\quad - D\left( {y\left( t \right)^{T} g_{{\mathop x\limits^{.}}} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right) + z\left( t \right)^{T} h_{{\mathop x\limits^{.}}} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right)} \right) = 0,\quad t \in I \\ & \bar{x}\left( t \right)^{T} B^{i} \left( t \right)w^{i} \left( t \right) = \left( {\bar{x}\left( t \right)^{T} B^{i} \left( t \right)\bar{x}\left( t \right)} \right)^{1/2} ,\left( {w^{i} \left( t \right)^{T} B^{i} \left( t \right)w^{i} \left( t \right)} \right)^{1/2} { \leqq }1,i \in K,\quad t \in I \\ & \left( {\bar{\lambda }^{1} ,\bar{\lambda }^{2} , \ldots ,\bar{\lambda }^{p} ,y\left( t \right)} \right){ \geqq }0,\quad t \in I \\ \end{aligned}$$

which give the required optimality conditions.

Theorem 2 (Karush - Kuhn - Tucker necessary optimality conditions):

Assume that

• (i) $\bar{x}\left( t \right)$ be an efficient solution of (NWVEP) and

• (ii) for each $r \in K,$ the constraints of (NWVEP_r) satisfy Slater’s [5] or Robinsons [5] condition at $\bar{x}\left( t \right) .$

Then there exist $\bar{\lambda } \in R^{p}$ and piecewise smooth functions $\bar{y} :\,I \to R^{m} ,$ $\bar{z} :\,I \to R^{k}$ and $w^{i} :\,I \to R^{m} ,\;i \in K$ such that

$$\begin{aligned} & \sum\limits_{i = 1}^{p} {\lambda^{i} \left( {f_{x}^{i} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right) + B^{i} \left( t \right)w^{i} \left( t \right) - Df_{{\mathop x\limits^{.}}}^{i} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right)} \right)} \\ &\quad + y\left( t \right)^{T} g_{x} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right) + z\left( t \right)^{T} h_{x} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right) \\ & \quad - D\left( {y\left( t \right)^{T} g_{{\mathop x\limits^{.}}} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right) + z\left( t \right)^{T} h_{{\mathop x\limits^{.}}} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right)} \right) = 0,\quad t \in I \\ & y\left( t \right)^{T} g\left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right) = 0,y\left( t \right){ \geqq }0,t \in I,\lambda > 0,\quad \sum\limits_{i = 1}^{p} {\lambda^{i} } = 1. \\ & \bar{x}\left( t \right)^{T} B^{i} \left( t \right)w^{i} \left( t \right) = \left( {\bar{x}\left( t \right)^{T} B^{i} \left( t \right)\bar{x}\left( t \right)} \right)^{1/2} ,\left( {w^{i} \left( t \right)^{T} B^{i} \left( t \right)w^{i} \left( t \right)} \right)^{1/2} { \leqq }1, \quad i \in K \\ \end{aligned} $$

Proof:

Since $\bar{x}\left( t \right)$ is an efficient solution of (NWVEP), by Lemma 1 $\bar{x}\left( t \right)$ is an optimal solution of (NWVEP_r), by the Karush-Kuhn-Tucker conditions given earlier, for each $r \in K,$ there exist $\bar{v}_{r}^{i} \in R,\,\,r \in K_{r}$ and piecewise smooth functions $\mu_{r}^{j} \in R,\,\,j = 1,2, \ldots ,m,$ $\delta_{r}^{l} \in R,\,\,l = 1,2, \ldots ,k$ such that

$$ \begin{aligned} & f_{x}^{r} (t,\bar{x},\dot{\bar{x}}) + B^{i} (t)w^{i} (t) - Df_{{\mathop x\limits^{.}}}^{r} (t,\bar{x},\dot{\bar{x}}) \\ &\quad + \sum\limits_{{i \in K_{r} }} {\bar{v}_{r}^{i} (f_{x}^{i} (t,\bar{x},\dot{\bar{x}}) + B^{i} (t)w^{i} (t) - Df_{{\mathop x\limits^{.}}}^{i} (t,\bar{x},\dot{\bar{x}}))} \\ & \sum\limits_{j = 1}^{m} {\left( {\mu_{r}^{j} g_{x}^{j} \left( {t,\bar{x},\dot{\bar{x}}} \right) - D\left( {\mu_{r}^{j} g_{{\mathop x\limits^{.}}}^{j} \left( {t,\bar{x},\dot{\bar{x}}} \right)} \right)} \right)} \\ & \quad + \sum\limits_{l = 1}^{k} {\left( {\delta_{r}^{l} h_{x}^{j} \left( {t,\bar{x},\dot{\bar{x}}} \right) - D\left( {\delta_{r}^{l} h_{{\mathop x\limits^{.}}}^{j} \left( {t,\bar{x},\dot{\bar{x}}} \right)} \right)} \right) = 0} \\ & \sum\limits_{j = 1}^{m} {\mu_{r}^{j} \left( t \right)g^{j} \left( {t,\bar{x},\dot{\bar{x}}} \right) = 0,} \bar{v}_{r}^{i} { \geqq }0,i \in K_{r} ,y\left( t \right){ \geqq }0,\quad t \in I \\ \end{aligned}$$

Summing over $i \in K ,$ we get

$$ \begin{aligned} & \sum\limits_{{i = 1}}^{p} {\left( {\bar{v}_{1}^{i} + \bar{v}_{2}^{i} + \cdots + \bar{v}_{p}^{i} } \right)\left( {f_{x}^{i} \left( {t,\bar{x},\dot{\bar{x}}} \right) + B^{i} \left( t \right)w^{i} \left( t \right) - Df_{{\mathop x\limits^{.}}}^{i} \left( {t,\bar{x},\dot{\bar{x}}} \right)} \right)} \\ & \quad + \sum\limits_{{j = 1}}^{m} {\left\{ {\left( {\mu _{1}^{j} \left( t \right) + \mu _{2}^{j} \left( t \right) + \cdots + \mu _{p}^{j} \left( t \right)} \right)g_{x}^{j} \left( {t,\bar{x},\dot{\bar{x}}} \right)} \right.} \\ & \quad \left. { - D\left( {\mu _{1}^{j} \left( t \right) + \mu _{2}^{j} \left( t \right) + \cdots + \mu _{p}^{j} \left( t \right)} \right)g_{{\mathop x\limits^{.}}}^{j} \left( {t,\bar{x},\dot{\bar{x}}} \right)} \right\} \\ & \quad + \sum\limits_{{l = 1}}^{k} {\left\{ {\left( {\delta _{1}^{l} \left( t \right) + \delta _{2}^{l} \left( t \right) + \cdots + \delta _{p}^{l} \left( t \right)} \right)h_{x}^{l} \left( {t,\bar{x},\dot{\bar{x}}} \right)} \right.} \\ & \quad \left. { - D\left( {\delta _{1}^{l} \left( t \right) + \delta _{2}^{l} \left( t \right) + \cdots + \delta _{p}^{l} \left( t \right)} \right)h_{{\mathop x\limits^{.}}}^{l} \left( {t,\bar{x},\dot{\bar{x}}} \right)} \right\} = 0 \\ & \sum\limits_{{j = 1}}^{m} {\left( {\mu _{1}^{j} \left( t \right) + \mu _{2}^{j} \left( t \right) + \cdots + \mu _{p}^{j} \left( t \right)} \right)g^{j} \left( {t,\bar{x},\dot{\bar{x}}} \right) = 0,} \quad t \in I \\ \end{aligned} $$

where $\bar{v}_{r}^{i} = 1$ for each $i \in K .$

Equivalently,

$$ \begin{aligned} & \sum\limits_{i = 1}^{k} {\bar{v}^{i} \left( {f_{x}^{i} \left( {t,\bar{x},\dot{\bar{x}}} \right) + B^{i} \left( t \right)w^{i} \left( t \right) - Df_{{\mathop x\limits^{.}}}^{i} \left( {t,\bar{x},\dot{\bar{x}}} \right)} \right)} \\ & \quad + \sum\limits_{j = 1}^{m} {\left\{ {\left( {\mu^{j} g_{x}^{j} \left( {t,\bar{x},\dot{\bar{x}}} \right)} \right) - D\left( {\mu^{j} g_{{\mathop x\limits^{.}}}^{j} \left( {t,\bar{x},\dot{\bar{x}}} \right)} \right)} \right\}} \\ & \quad + \sum\limits_{l = 1}^{k} {\left( {\delta^{l} h_{x}^{j} \left( {t,\bar{x},\dot{\bar{x}}} \right) - D\left( {\delta^{l} h_{{\mathop x\limits^{.}}}^{j} \left( {t,\bar{x},\dot{\bar{x}}} \right)} \right)} \right) = 0} \\ \end{aligned} $$

where $\bar{v}^{i} = 1 + \sum\limits_{{r \in K_{r} }} {v_{r}^{i} } > 0,\;i \in K,$ $\mu^{j} \left( t \right) = \sum\limits_{r = 1}^{k} {\mu_{r}^{j} \left( t \right)} \geqq 0,t \in I, j = 1,2, \ldots ,m$ and

$$\begin{aligned} & \delta ^{l} \left( t \right) = \sum\limits_{{r = 1}}^{k} {\delta _{r}^{l} \left( t \right)} { \geqq }0,t \in I,\quad l = 1,2, \ldots ,k. \\ & \quad \quad \; \Rightarrow \lambda ^{i} = \frac{{\bar{v}^{i} }}{{\sum\limits_{{r = 1}}^{p} {\bar{v}^{i} } }},\quad i \in K, \\ & y^{j} \left( t \right) = \frac{{\mu ^{j} \left( t \right)}}{{\sum\limits_{{r = 1}}^{p} {\bar{v}^{i} } }},\quad j = 1,2, \ldots ,m, \\ & z^{l} \left( t \right) = \frac{{\delta ^{l} \left( t \right)}}{{\sum\limits_{{r = 1}}^{p} {\bar{v}^{i} } }},\quad l = 1,2, \ldots ,k. \\ \end{aligned}$$

We get

$$\begin{aligned} & \sum\limits_{i = 1}^{p} {\lambda^{i} \left( {f_{x}^{i} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right) + B^{i} \left( t \right)w^{i} \left( t \right) - Df_{{\mathop x\limits^{.}}}^{i} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right)} \right)} \\ &\quad + \sum\limits_{j = 1}^{m} {\left( {y^{j} \left( t \right)g_{x}^{j} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right) - D\left( {y^{j} \left( t \right)g_{{\mathop x\limits^{.}}}^{j} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right)} \right)} \right)} \\ & \quad + \sum\limits_{l = 1}^{k} {\left( {z^{l} \left( t \right)h_{x}^{l} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right) - D\left( {z^{l} \left( t \right)h_{x}^{l} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right)} \right)} \right)} = 0 \\ & \sum\limits_{j = 1}^{m} {y^{j} \left( t \right)g^{j} \left( {t,\bar{x},\dot{\bar{x}}} \right) = 0,\quad t \in I} \\ \end{aligned} $$

or

$$ \begin{aligned} & \sum {\lambda^{i} \left( {f_{x}^{i} \left( {t,\bar{x},\dot{\bar{x}}} \right) + B^{i} \left( t \right)w^{i} \left( t \right) - Df_{{\mathop x\limits^{.}}}^{i} \left( {t,\bar{x},\dot{\bar{x}}} \right)} \right)} \\ & \quad + \left( {y\left( t \right)^{T} g_{x} \left( {t,\bar{x},\dot{\bar{x}}} \right) + z\left( t \right)^{T} h_{x} \left( {t,\bar{x},\dot{\bar{x}}} \right)} \right) \\ & \quad - D\left( {y\left( t \right)^{T} g_{{\mathop x\limits^{.}}} \left( {t,\bar{x},\dot{\bar{x}}} \right) + z\left( t \right)^{T} h_{{\mathop x\limits^{.}}} \left( {t,\bar{x},\dot{\bar{x}}} \right)} \right) = 0,\quad t \in I \\ & y\left( t \right)^{T} g\left( {t,\bar{x},\dot{\bar{x}}} \right) = 0,y\left( t \right) = \left( {y^{1} \left( t \right),y^{2} \left( t \right), \ldots ,y^{m} \left( t \right)} \right)\geqq 0,\quad t \in I \\ & \lambda = \left( {\lambda^{1} ,\lambda^{2} , \ldots ,\lambda^{k} } \right) > 0,\quad \sum\limits_{i = 1}^{k} {\lambda^{i} } = 1 \\ \end{aligned} $$

4 Wolfe Type Second-Order Duality

We construct the following problem as the dual to the problem (P):

$$\begin{array}{*{20}l} {( {\mathbf{D}} ){\text{:}}} & {{\text{Maximize}}} \\ {} & {\begin{array}{*{20}l} {\left( {\int\limits_{I} {\left( {f^{1} \left( {t,u\left( t \right),\dot{u}\left( t \right)} \right) + u\left( t \right)^{T} B^{1} \left( t \right)w^{1} \left( t \right) + y\left( t \right)^{T} g\left( {t,u\left( t \right),\dot{u}\left( t \right)} \right)} \right.} } \right.} \hfill \\ {\left. {\quad + z\left( t \right)^{T} h\left( {t,u\left( t \right),\dot{u}\left( t \right)} \right) - \frac{1}{2}\beta \left( t \right)^{T} H^{1} \beta \left( t \right)} \right)dt, \ldots ,} \hfill \\ {\int\limits_{I} {\left( {f^{p} \left( {t,u\left( t \right),\dot{u}\left( t \right)} \right) + u\left( t \right)^{T} B^{p} \left( t \right)w^{p} \left( t \right) + y\left( t \right)^{T} g\left( {t,u\left( t \right),\dot{u}\left( t \right)} \right)} \right.} } \hfill \\ {\left. {\left. {\quad + z\left( t \right)^{T} h\left( {t,u\left( t \right),\dot{u}\left( t \right)} \right) - \frac{1}{2}\beta \left( t \right)^{T} H^{p} \beta \left( t \right)} \right)dt} \right)} \hfill \\ \end{array} } \\ \end{array} $$

subject to

$$ u\left( a \right) = 0 = u\left( b \right) $$

(4)

$$\begin{aligned} & \sum\limits_{{i = 1}}^{p} {\lambda ^{i} } \left( {f_{u}^{i} \left( {t,u\left( t \right),\dot{u}\left( t \right)} \right) + B^{i} \left( t \right)w^{i} \left( t \right) - D f_{u}^{i} \left( {t,u\left( t \right),\dot{u}\left( t \right)} \right)} \right) \\ & \quad + y\left( t \right)^{T} g_{u} \left( {t,u\left( t \right),\dot{u}\left( t \right)} \right)n + z\left( t \right)^{T} h_{u} \left( {t,u\left( t \right),\dot{u}\left( t \right)} \right) \\ & \quad - D\left( {y\left( t \right)^{T} g_{{\dot{u}}} \left( {t,u\left( t \right),\dot{u}\left( t \right)} \right) + z\left( t \right)^{T} h_{{\dot{u}}} \left( {t,u\left( t \right),\dot{u}\left( t \right)} \right)} \right) \\ & \quad + H\beta \left( t \right) = 0,\;\;\;{\kern 1pt} t \in I \\ \end{aligned}$$

(5)

$$ w^{i} \left( t \right)^{T} B^{i} \left( t \right)w^{i} \left( t \right){ \leqq }1,\quad i \in K $$

(6)

$$ y\left( t \right){ \geqq }0,\quad t \in I $$

(7)

$$ \lambda > 0,\quad \sum\limits_{i = 1}^{p} {\lambda^{i} = 1} $$

(8)

where

$$ \begin{aligned} H^{i} & = f_{xx}^{i} - 2Df_{{x\dot{x}}}^{i} + D^{2} f_{{\dot{x}\dot{x}}}^{i} - D^{3} f_{{\dot{x}\ddot{\textit{x}}}}^{i} + \left({y\left(t \right)^{T} g_{x} + z\left(t \right)^{T} h_{x}} \right)_{x} - 2D\left({y\left(t \right)^{T} g_{x} + z\left(t \right)^{T} h_{x}} \right)_{{\mathop x\limits^{.}}} \\ & \quad + D^{2} \left({y\left(t \right)^{T} g_{{\mathop x\limits^{.}}} + z\left(t \right)^{T} h_{{\mathop x\limits^{.}}}} \right)_{{\mathop x\limits^{.}}} - D^{3} \left({y\left(t \right)^{T} g_{{\mathop x\limits^{.}}} + z\left(t \right)^{T} h_{{\mathop x\limits^{.}}}} \right)_{{\ddot{\textit{x}}}} \\ H & = \lambda^{T} f_{xx} - 2D\lambda^{T} f_{{x\dot{x}}} + D^{2} \lambda^{T} f_{{\dot{x}\dot{x}}} - D^{3} \lambda^{T} f_{{\dot{x}\ddot{\textit{x}}}} + \left({y\left(t \right)^{T} g_{x} + z\left(t \right)^{T} h_{x}} \right)_{x} \\ & \quad - 2D\left({y\left(t \right)^{T} g_{x} + z\left(t \right)^{T} h_{x}} \right)_{{\mathop x\limits^{.}}} + D^{2} \left({y\left(t \right)^{T} g_{{\mathop x\limits^{.}}} + z\left(t \right)^{T} h_{{\mathop x\limits^{.}}}} \right)_{{\mathop x\limits^{.}}} - D^{3} \left({y\left(t \right)^{T} g_{{\mathop x\limits^{.}}} + z\left(t \right)^{T} h_{{\mathop x\limits^{.}}}} \right)_{{\ddot{\textit{x}}}}. \\ \end{aligned} $$

Theorem 3 (Weak Duality):

Let $x\left( t \right) \in C_{p}$ and $\left( {u\left( t \right),\lambda ,y\left( t \right),z\left( t \right),w^{1} \left( t \right), \ldots ,w^{p} \left( t \right)} \right) \in C_{D}$ such that for $\int\limits_{I} {\left( {f^{i} \left( {t,.,.} \right) + \left( . \right)^{T} B^{i} \left( t \right)w^{i} \left( t \right) + y\left( t \right)^{T} g\left( {t,.,.} \right) + z\left( t \right)^{T} h\left( {t,.,.} \right)} \right)dt,} i \in K,$ is second-order pseudoinvex for all $w^{i} \left( t \right) \in R^{n} ,\,i \in K$ with respect to $\eta .$ Then

$$ \begin{aligned} & \int\limits_{I} {\left( {f^{i} \left( {t,x\left( t \right),\dot{x}\left( t \right)} \right) + \,\left( {x\left( t \right)^{T} B^{i} \left( t \right)x\left( t \right)} \right)^{{1/2}} } \right)} \,dt \\ & \quad < \int\limits_{I} {\left( {f^{i} \left( {t,u\left( t \right),\dot{u}\left( t \right)} \right) + u\left( t \right)^{T} B^{i} \left( t \right)w^{i} \left( t \right) + y\left( t \right)^{T} g\left( {t,u\left( t \right),\dot{u}\left( t \right)} \right)} \right.} \\ & \quad \quad+ \left. { z\left( t \right)^{T} h\left( {t,u\left( t \right),\dot{u}\left( t \right)} \right) - \frac{1}{2}\beta \left( t \right)^{T} H^{i} \beta \left( t \right)} \right)dt,i \in K = \left\{ {1,2,...,p} \right\} \\ \end{aligned} $$

and

$$ \begin{aligned} & \int\limits_{I} {\left( {f^{j} \left( {t,x\left( t \right),\dot{x}\left( t \right)} \right) + \left( {x\left( t \right)^{T} B^{j} \left( t \right)x\left( t \right)} \right)^{1/2} } \right)} dt \\ & \quad \leqq \int\limits_{I} {\left( {f^{j} \left( {t,u\left( t \right),\dot{u}\left( t \right)} \right) + u\left( t \right)^{T} B^{j} \left( t \right)w^{j} \left( t \right) + y\left( t \right)^{T} g\left( {t,u\left( t \right),\dot{u}\left( t \right)} \right)} \right.} \\ & \quad \quad + \left. {z\left( t \right)^{T} h\left( {t,u\left( t \right),\dot{u}\left( t \right)} \right) - \frac{1}{2}\beta \left( t \right)^{T} H^{j} \beta \left( t \right)} \right)dt,\quad {\text {for\,all}}\;j \in K,\;j \ne i. \\ \end{aligned}$$

cannot hold.

Proof:

Suppose to the contrary that there is $\bar{x}\left( t \right)$ feasible for (VP) and $\left( {u\left( t \right),\lambda ,y\left( t \right),z\left( t \right),w^{1} \left( t \right), \ldots ,w^{p} \left( t \right)} \right)$ feasible for (DV) such that

$$ \begin{aligned} & \int\limits_{I} {\left( {f^{i} \left( {t,x\left( t \right),\dot{x}\left( t \right)} \right) + \left( {x\left( t \right)^{T} B^{i} \left( t \right)x\left( t \right)} \right)^{{1/2}} } \right)} dt \\ & \quad < \int\limits_{I} {\left( {f^{i} \left( {t,u\left( t \right),\dot{u}\left( t \right)} \right) + u\left( t \right)^{T} B^{i} \left( t \right)w^{i} \left( t \right) + y\left( t \right)^{T} g\left( {t,u\left( t \right),\dot{u}\left( t \right)} \right)} \right.} \\ & \quad \quad + \left. {z\left( t \right)^{T} h\left( {t,u\left( t \right),\dot{u}\left( t \right)} \right) - \frac{1}{2}\beta \left( t \right)^{T} H^{i} \beta \left( t \right)} \right)dt,\;\;\;{\kern 1pt} {\text{for}}\;{\text{some}}\;i \in \left\{ {1,2, \ldots ,p} \right\} \\ \end{aligned}$$

and

$$\begin{aligned} & \int\limits_{I} {\left( {f^{j} \left( {t,x\left( t \right),\dot{x}\left( t \right)} \right) + \,\left( {x\left( t \right)^{T} B^{j} \left( t \right)x\left( t \right)} \right)^{1/2} } \right)} dt \\ &\quad \leqq \int\limits_{I} {\left( {f^{j} \left( {t,u\left( t \right),\dot{u}\left( t \right)} \right) + u\left( t \right)^{T} B^{j} \left( t \right)w^{j} \left( t \right) + y\left( t \right)^{T} g\left( {t,u\left( t \right),\dot{u}\left( t \right)} \right)} \right.} \\ & \quad \quad + \left. { z\left( t \right)^{T} h\left( {t,u\left( t \right),\dot{u}\left( t \right)} \right) - \frac{1}{2}\beta \left( t \right)^{T} H^{j} \beta \left( t \right)} \right)dt,\quad \text{for} \, \text{all}\;j \in K,\;j \ne i. \\ \end{aligned} $$

Then, using $x\left( t \right)^{T} B^{i} \left( t \right)w^{i} \left( t \right) \le \left( {x\left( t \right)^{T} B^{i} \left( t \right)x\left( t \right)} \right)^{1/2} \left( {w^{i} \left( t \right)^{T} B^{i} \left( t \right)w^{i} \left( t \right)} \right)^{1/2} ,\quad i \in K$

We have

$$ \begin{aligned} & \int\limits_{I} {\left( {f^{i} \left( {t,x\left( t \right),\dot{x}\left( t \right)} \right) + x\left( t \right)^{T} \,B^{i} \left( t \right)w^{i} \left( t \right) + \,y\left( t \right)^{T} g\left( {t,x\left( t \right),\dot{x}\left( t \right)} \right) + z\left( t \right)^{T} h\left( {t,x\left( t \right),\dot{x}\left( t \right)} \right)\,} \right)\,} dt \\ & \quad < \int\limits_{I} {\left( {f^{i} \left( {t,u\left( t \right),\dot{u}\left( t \right)} \right) + u\left( t \right)^{T} B^{i} \left( t \right)w^{i} \left( t \right) + y\left( t \right)^{T} g\left( {t,u\left( t \right),\dot{u}\left( t \right)} \right)} \right.} \\ & \quad \quad + \left. { z\left( t \right)^{T} h\left( {t,u\left( t \right),\dot{u}\left( t \right)} \right) - \frac{1}{2}\beta \left( t \right)^{T} H^{i} \beta \left( t \right)} \right)dt,\,\,i \in K \\ & \int\limits_{I} {\left( {f^{j} \left( {t,x\left( t \right),\dot{x}\left( t \right)} \right) + x\left( t \right)^{T} \,B^{j} \left( t \right)w^{j} \left( t \right) + \,y\left( t \right)^{T} g\left( {t,x\left( t \right),\dot{x}\left( t \right)} \right) + z\left( t \right)^{T} h\left( {t,x\left( t \right),\dot{x}\left( t \right)} \right)\,} \right)\,} dt \\ & \quad \leqq \int\limits_{I} {\left( {f^{j} \left( {t,u\left( t \right),\dot{u}\left( t \right)} \right) + u\left( t \right)^{T} B^{j} \left( t \right)w^{j} \left( t \right) + y\left( t \right)^{T} g\left( {t,u\left( t \right),\dot{u}\left( t \right)} \right)} \right.} \\ & \quad \quad + \left. { z\left( t \right)^{T} h\left( {t,u\left( t \right),\dot{u}\left( t \right)} \right) - \frac{1}{2}\beta \left( t \right)^{T} H^{j} \beta \left( t \right)} \right)dt,\quad \text{for}\;\text{all}\;\text{j}\;\text{with}\;j \ne i. \\ \end{aligned}$$

Since $\int\limits_{I} {\left( {f^{i} \left( {t,.,.} \right) + \left( . \right)^{T} B^{i} \left( t \right)w^{i} \left( t \right) + y\left( t \right)^{T} g\left( {t,.,.} \right) + z\left( t \right)^{T} h\left( {t,.,.} \right)} \right)dt,}$ is second-order pseudoinvex for all $i \in K,$ then

$$ \begin{aligned} & \int\limits_{I} {\left[ {\eta^{T} \left( {f_{u}^{i} + B^{i} \left( t \right)w^{i} \left( t \right) + y\left( t \right)^{T} g_{u} \left( {t,u\left( t \right),\dot{u}\left( t \right)} \right) + z\left( t \right)^{T} h_{u} \left( {t,u\left( t \right),\dot{u}\left( t \right)} \right)} \right)} \right.} \\ & \quad \left. { + \,\left( {\frac{d\eta }{dt}} \right)^{T} \left( {f_{{\dot{u}}}^{i} + \,y\left( t \right)^{T} g_{{\dot{u}}} + z\left( t \right)^{T} h_{{\dot{u}}} } \right) + \eta^{T} H^{i} \beta \left( t \right)} \right] < 0,\quad for\,all\,i \in K, \\ \end{aligned} $$

and

$$\begin{aligned} & \int\limits_{I} {\left[ {\eta^{T} \left( {f_{u}^{j} + B^{j} \left( t \right)w^{j} \left( t \right) + y\left( t \right)^{T} g_{u} \left( {t,u\left( t \right),\dot{u}\left( t \right)} \right) + z\left( t \right)^{T} h_{u} \left( {t,u\left( t \right),\dot{u}\left( t \right)} \right)} \right)} \right.} \\ & \quad + \left. {\left( {\frac{d\eta }{dt}} \right)^{T} \left( {f_{{\dot{u}}}^{j} + \,y\left( t \right)^{T} g_{{\dot{u}}} + z\left( t \right)^{T} h_{{\dot{u}}} } \right) + \eta^{T} H^{j} \beta \left( t \right)} \right]\leqq \,0,\quad j \ne i. \\ \end{aligned} $$

Thus

$$ \begin{aligned} 0 > & \int\limits_{I} {\left[ {\eta^{T} \left( {\sum\limits_{i = 1}^{p} {\lambda^{i} \left( {f_{u}^{i} + B^{i} \left( t \right)w^{i} \left( t \right)} \right) + y\left( t \right)^{T} g_{u} \left( {t,u\left( t \right),\dot{u}\left( t \right)} \right) + z\left( t \right)^{T} h_{u} \left( {t,u\left( t \right),\dot{u}\left( t \right)} \right)} } \right)} \right.} \\ & + \left. { \left( {\frac{d\eta }{dt}} \right)^{T} \left( {\lambda^{T} f_{{\dot{u}}} + \,y\left( t \right)^{T} g_{{\dot{u}}} + z\left( t \right)^{T} h_{{\dot{u}}} } \right) + \eta^{T} H\,\beta \left( t \right)} \right]dt \\ = & \int\limits_{I} {\left[ {\eta^{T} \left( {\sum\limits_{i = 1}^{p} {\lambda^{i} \left( {f_{u}^{i} + \,B^{i} \left( t \right)w^{i} \left( t \right)} \right) + y\left( t \right)^{T} g_{u} \left( {t,u\left( t \right),\dot{u}\left( t \right)} \right) + z\left( t \right)^{T} h_{u} \left( {t,u\left( t \right),\dot{u}\left( t \right)} \right)} } \right)} \right]} dt \\ & \left. { - D\left( {\lambda^{T} f_{{\dot{u}}} + \,y\left( t \right)^{T} g_{{\dot{u}}} + z\left( t \right)^{T} h_{{\dot{u}}} } \right) + H\,\beta \left( t \right)} \right]dt + \eta^{T} \left( {\lambda^{T} f_{{\dot{u}}} + \,y\left( t \right)^{T} g_{{\dot{u}}} + z\left( t \right)^{T} h_{{\dot{u}}} } \right)\left| \begin{gathered} t = b \hfill \\ t = a \hfill \\ \end{gathered} \right. \\ \end{aligned} $$

Using $\eta = 0,$ at t = a and t = b, we obtain,

$$ \begin{aligned} \Rightarrow & \int\limits_{I} {\eta^{T} } \left[ {\sum\limits_{i = 1}^{p} {\lambda^{i} \left( {f_{u}^{i} + \,B^{i} \left( t \right)w^{i} \left( t \right)} \right) + y\left( t \right)^{T} g_{u} \left( {t,u\left( t \right),\dot{u}\left( t \right)} \right) + z\left( t \right)^{T} h_{u} \left( {t,u\left( t \right),\dot{u}\left( t \right)} \right)} } \right. \\ & \left. { - D\left( {\lambda^{T} f_{{\dot{u}}} + y\left( t \right)^{T} g_{{\dot{u}}} + z\left( t \right)^{T} h_{{\dot{u}}} } \right) + H\beta \left( t \right)} \right]dt < 0 \\ \end{aligned} $$

contradicting the constraint of (NWVED). Thus the validity of the conclusion of the theorem follows.

Theorem 4 (Strong Duality):

Let $\bar{x}\left( t \right)$ be efficient and normal solution for (NWVEP), then there exist $\lambda \in R^{k}$ and piecewise smooth $y :I \to R^{m} ,$ $z :I \to R^{l} ,$ and $w^{i} :I \to R^{n} ,$ $i \in K$ such that $\left( {\bar{x},\lambda ,y,z,w^{1} , \ldots ,w^{p} , \beta = 0} \right)$ is feasible for (WVED) and the two objective functionals are equal. Furthermore, if the hypothesis of theorem hold, then $\left( {\bar{x},\lambda ,y,z,w^{1} , \ldots ,w^{p} ,\beta } \right)$ is efficient for the problem (NWVED).

Proof:

By Theorem 2, there exist $\lambda = \left( {\lambda^{1} ,\lambda^{2} , \ldots ,\lambda^{p} } \right) \in R^{p}$ and piecewise smooth $y :I \to R^{m} ,$ $z :I \to R^{l}$ and $w^{i} :I \to R^{l} , i = 1,2, \ldots ,p$ such that

$$ \begin{aligned} &\sum\limits_{{i = 1}}^{p} {\lambda ^{i} } \left( {f_{x}^{i} \left( {t,\bar{x},\dot{\bar{x}}} \right) + \bar{x}\left( t \right)B^{i} \left( t \right)w^{i} \left( t \right) - D\,f_{{\mathop x\limits^{.}}}^{i} \left( {t,\bar{x},\dot{\bar{x}}} \right)} \right)\, \hfill \\ & \quad + y\left( t \right)^{T} g_{x} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right) + z\left( t \right)^{T} h_{x} \left( {t,x\left( t \right),\dot{x}\left( t \right)} \right) \hfill \\ & \quad - D\left( {y\left( t \right)^{T} g_{{\mathop x\limits^{.}}} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right) + z\left( t \right)^{T} h_{{\mathop x\limits^{.}}} \left( {t,x\left( t \right),\dot{x}\left( t \right)} \right)} \right) = 0, \quad t \in I \hfill \\ & {\text{ }}y\left( t \right)^{T} g\left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right) = 0,y\left( t \right){ \geqq }0,t \in I,\lambda {\text{ > }}0, \quad \sum\limits_{{i = 1}}^{p} {\lambda ^{i} = 1} . \hfill \\& \bar{x}\left( t \right)^{T} B^{i} \left( t \right)w^{i} \left( t \right) = \left( {\bar{x}\left( t \right)^{T} B^{i} \left( t \right)\bar{x}\left( t \right)} \right)^{{1/2}} , w^{i} \left( t \right)^{T} B^{i} \left( t \right)w^{i} \left( t \right){ \leqq }1,\quad t \in I,i \in K \hfill \\ \end{aligned} $$

Thus $\left( {\bar{x},\lambda ,y,z,w^{1} , \ldots ,w^{p} , \beta = 0} \right)$ is feasible for (NWVED) and for all $i \in K.$

$$ \begin{aligned} & \int\limits_{I} {\left( {f^{i} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right) + \,\left( {\bar{x}\left( t \right)^{T} B^{i} \left( t \right)\bar{x}\left( t \right)} \right)^{1/2} } \right)} dt \\ & \quad < \int\limits_{I} {\left( {f^{i} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right) + \bar{x}\left( t \right)^{T} B^{i} \left( t \right)\bar{w}^{i} \left( t \right)} \right.} \\ & \quad \quad + \left. {y\left( t \right)^{T} g\left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right) + z\left( t \right)^{T} h\left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right)} \right)dt,\quad t \in I,i \in K \\ \end{aligned} $$

This implies that, the objective functional values are equal.

If $\left( {\bar{x},\lambda ,y,z,w^{1} ,w^{2} , \ldots ,w^{p} , \beta = 0} \right)$ is not efficient solution of (NWVED), then there exists feasible $\left( {u^{*} ,\lambda^{*} ,y^{*} ,z^{*} ,w^{1} ,, \ldots ,w^{p} ,\beta^{*} } \right)$ for (NWVED) such that

$$\begin{aligned} & \int\limits_{I} {\left( {f^{i} \left( {t,u^{*} ,\dot{u}^{*} } \right) + u^{*} \left( t \right)^{T} B^{i} \left( t \right)w^{i} \left( t \right) + y^{*} \left( t \right)^{T} g\left( {t,u^{*} ,\dot{u}^{*} } \right)} \right.} \\ & \quad + \left. { z^{*} \left( t \right)^{T} h\left( {t,u^{*} ,\dot{u}^{*} } \right) - \frac{1}{2}\beta ^{*} \left( t \right)^{T} H^{i} \beta ^{*} \left( t \right)} \right)dt \\ & \quad > \int\limits_{I} {\left( {f^{i} \left( {t,\bar{x},\dot{\bar{x}}} \right) + \bar{x}\left( t \right)^{T} B^{i} \left( t \right)w^{i} \left( t \right) + y^{*} \left( t \right)^{T} g\left( {t,\bar{x},\dot{\bar{x}}} \right)} \right.} \\ & \quad \quad + \left. { z^{*} \left( t \right)^{T} h\left( {t,\bar{x},\dot{\bar{x}}} \right) - \frac{1}{2}\beta ^{*} \left( t \right)^{T} H^{i} \beta ^{*} \left( t \right)} \right)dt,\quad i \in K. \\ \end{aligned} $$

and

$$ \begin{aligned} & \int\limits_{I} {\left( {f^{j} \left( {t,u^{*} ,\dot{u}^{*} } \right) + u^{*} \left( t \right)^{T} B^{j} \left( t \right)w^{i} \left( t \right) + y^{*} \left( t \right)^{T} g\left( {t,u^{*} ,\dot{u}^{*} } \right)} \right.} \\ & \quad + \left. { z^{*} \left( t \right)^{T} h\left( {t,u^{*} ,\dot{u}^{*} } \right) - \frac{1}{2}\beta ^{*} \left( t \right)^{T} H^{*j} \beta ^{*} \left( t \right)} \right)dt \\ & \quad \geqq \int\limits_{I} {\left( {f^{j} \left( {t,\bar{x},\dot{\bar{x}}} \right) + \bar{x}\left( t \right)^{T} B^{j} \left( t \right)w^{j} \left( t \right) + y^{*} \left( t \right)^{T} g\left( {t,\bar{x},\dot{\bar{x}}} \right)} \right.} \\ & \quad \quad + \left. { z^{*} \left( t \right)^{T} h\left( {t,\bar{x},\dot{\bar{x}}} \right) - \frac{1}{2}\beta ^{*} \left( t \right)^{T} H^{j} \beta ^{*} \left( t \right)} \right)dt,\quad j \ne i \\ \end{aligned} $$

Since $\int\limits_{I} {\left( {f^{i} \left( {t,.,.} \right) + \left( . \right)^{T} B^{i} \left( t \right)w^{i} \left( t \right) + y\left( t \right)^{T} g\left( {t,.,.} \right) + z\left( t \right)^{T} h\left( {t,.,.} \right)} \right)dt,}$ is second-order pseudoinvex with respect to $\eta$, as earlier

$$\begin{aligned} & \int\limits_{I} {\eta ^{T} } \left[ {\left( {f_{u}^{i} \left( {t,u^{*} ,\dot{u}^{*} } \right) + \,B^{i} \left( t \right)w^{*i} - Df_{{\dot{u}}}^{i} \left( {t,u^{*} ,\dot{u}^{*} } \right)} \right)} \right. \\ & \quad + y^{*} \left( t \right)^{T} g_{u} \left( {t,u^{*} ,\dot{u}^{*} } \right) + z^{*} \left( t \right)^{T} h_{u} \left( {t,u^{*} ,\dot{u}^{*} } \right) \\ & \quad \left. { - D\left( {y^{*} \left( t \right)^{T} g_{u} \left( {t,u^{*} ,\dot{u}^{*} } \right) + z^{*} \left( t \right)^{T} h_{u} \left( {t,u^{*} ,\dot{u}^{*} } \right) + H^{*} \beta \left( t \right)} \right)} \right]dt\,\, < \,\,0,\quad i \in K \\ & \int\limits_{I} {\eta ^{T} } \left[ {\left( {f_{u}^{j} \left( {t,u^{*} ,\dot{u}^{*} } \right) + \,B^{j} \left( t \right)w^{*i} - Df_{{\dot{u}}}^{j} \left( {t,u^{*} ,\dot{u}^{*} } \right)} \right)} \right. \\ & \quad + y^{*} \left( t \right)^{T} g_{u} \left( {t,u^{*} ,\dot{u}^{*} } \right) + z^{*} \left( t \right)^{T} h_{u} \left( {t,u^{*} ,\dot{u}^{*} } \right) \\ & \quad \left. { - D\left( {y^{*} \left( t \right)^{T} g_{u} \left( {t,u^{*} ,\dot{u}^{*} } \right) + z^{*} \left( t \right)^{T} h_{u} \left( {t,u^{*} ,\dot{u}^{*} } \right) + H^{*} \beta \left( t \right)} \right)} \right]dt\,\leqq\,0,\quad j \ne i. \\ \end{aligned} $$

Thus $\begin{aligned} & \int\limits_{I} {\eta^{T} } \left[ {\sum\limits_{i = 1}^{p} {\lambda^{i} } } \right.\left( {f_{u}^{i} \left( {t,u^{*} ,\dot{u}^{*} } \right) + \,B^{i} \left( t \right)w^{*i} - Df_{{\dot{u}}}^{i} \left( {t,u^{*} ,\dot{u}^{*} } \right)} \right) + y^{*} \left( t \right)^{T} g_{u} \left( {t,u^{*} ,\dot{u}^{*} } \right) \\ & \quad + z^{*} \left( t \right)^{T} h\left( {t,u^{*} ,\dot{u}^{*} } \right) \left. { - D\left( {y^{*} \left( t \right)^{T} g_{{\dot{u}}} \left( {t,u^{*} ,\dot{u}^{*} } \right) + z^{*} \left( t \right)^{T} h_{{\dot{u}}} \left( {t,u^{*} ,\dot{u}^{*} } \right) + H^{*} \beta \left( t \right)} \right)} \right]dt < 0 \\ \end{aligned}$ contradicting the feasibility of $\left( {u^{*} ,\lambda^{*} ,y^{*} ,z^{*} ,w^{*1} ,w^{*2} , \ldots ,w^{*p} ,\beta^{*} } \right)$ for (NWVED). Thus $\left( {\bar{x},\lambda ,y,z,w^{1} ,w^{2} , \ldots ,w^{p} , \beta = 0} \right)$ is efficient for the dual (NWVED).

Below is the Mangasarian [1] type Strict-Converse duality theorem:

Theorem 5 (Strict - Converse duality):

Let $\bar{x}\left( t \right)$ and $\left( {\bar{u}\left( t \right),\lambda ,y\left( t \right),z\left( t \right),w^{1} \left( t \right), \ldots ,w^{p} \left( t \right)} \right)$ be efficient solutions for the problems (NWVEP) and (NWVED) such that

$$ \begin{aligned} & \sum\limits_{i = 1}^{p} {\lambda^{i} } \int\limits_{I} {\left( {f^{i} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right) + \bar{x}\left( t \right)^{T} B^{i} \left( t \right)w^{i} \left( t \right)} \right)dt} \\ & \quad = \int\limits_{I} {\left( {\sum\limits_{i = 1}^{p} {\lambda^{i} } \left( {f^{i} \left( {t,\bar{u}\left( t \right),\dot{\bar{u}}\left( t \right)} \right) + \bar{u}\left( t \right)^{T} B^{i} \left( t \right)w^{i} \left( t \right)} \right)} \right.} \\ & \quad \quad + \left. {y\left( t \right)^{T} g\left( {t,\bar{u}\left( t \right),\dot{\bar{u}}\left( t \right)} \right) + z\left( t \right)^{T} h\left( {t,\bar{u}\left( t \right),\dot{\bar{u}}\left( t \right)} \right) - \frac{1}{2}\beta \left( t \right)^{T} H\beta \left( t \right)} \right)dt \\ \end{aligned} $$

(9)

If $\int\limits_{I} {\left( {\sum\limits_{i = 1}^{p} {\lambda^{i} } \left( {f^{i} \left( {t,.,.} \right) + \left( . \right)^{T} B^{i} \left( t \right)w^{i} \left( t \right) + y\left( t \right)^{T} g\left( {t,.,.} \right) + z\left( t \right)^{T} h\left( {t,.,.} \right)} \right)} \right.} dt$ is second-order strictly pseudoinvex with respect to $\eta$, then $\bar{x}\left( t \right) = \bar{u}\left( t \right), t \in I .$

Proof:

Suppose $\bar{x}\left( t \right) \ne \bar{u}\left( t \right),$ for $t \in I.$ By second-order strict pseudoinvexity with respect to $\eta ,$ (9) yields,

$$ \begin{aligned} 0 > & \int\limits_{I} {\left\{ {\eta^{T} \left( {\sum\limits_{i = 1}^{p} {\lambda^{i} \left( {f_{u}^{i} \left( {t,\bar{u}\left( t \right),\dot{\bar{u}}\left( t \right)} \right) + B^{i} \left( t \right)w^{i} \left( t \right) - Df_{{\dot{u}}}^{i} \left( {t,\bar{u}\left( t \right),\dot{\bar{u}}\left( t \right)} \right)} \right.} } \right.} \right.} \\ & \left. { + y\left( t \right)^{T} g_{u} \left( {t,\bar{u}\left( t \right),\dot{\bar{u}}\left( t \right)} \right) + z\left( t \right)^{T} h_{u} \left( {t,\bar{u}\left( t \right),\dot{\bar{u}}\left( t \right)} \right)} \right) \\ & \left. { + \left( {\frac{d\eta }{dt}} \right)^{T} \left( {\lambda f_{{\dot{u}}}^{i} + y\left( t \right)^{T} g_{{\dot{u}}} + z\left( t \right)^{T} h_{{\dot{u}}} \left( {t,u,\dot{u}} \right)} \right) + \eta^{T} H\beta \left( t \right)} \right\}dt \\ = & \int\limits_{I} {\eta^{T} \left\{ {\sum\limits_{i = 1}^{p} {\lambda^{i} \left( {f_{u}^{i} \left( {t,\bar{u}\left( t \right),\dot{\bar{u}}\left( t \right)} \right) + B^{i} \left( t \right)w^{i} \left( t \right) - Df_{{\dot{u}}}^{i} \left( {t,\bar{u}\left( t \right),\dot{\bar{u}}\left( t \right)} \right)} \right.} } \right.} \\ & \left. { + y\left( t \right)^{T} g_{u} \left( {t,\bar{u}\left( t \right),\dot{\bar{u}}\left( t \right)} \right) + z\left( t \right)^{T} h_{u} \left( {t,\bar{u}\left( t \right),\dot{\bar{u}}\left( t \right)} \right)} \right) \\ & \left. { - D\left( {f_{{\dot{u}}}^{i} + y\left( t \right)^{T} g_{{\dot{u}}} + z\left( t \right)^{T} h_{{\dot{u}}} \left( {t,u,\dot{u}} \right)} \right) + H\beta \left( t \right)} \right\}dt \\ & \left. { + \eta^{T} \left( {\lambda f_{{\dot{u}}}^{i} + y\left( t \right)^{T} g_{{\dot{u}}} + z\left( t \right)^{T} h_{{\dot{u}}} \left( {t,u,\dot{u}} \right)} \right)} \right|\begin{array}{*{20}c} {t = b} \\ {t = a} \\ \end{array} \\ \end{aligned} $$

Using $\eta = 0,$ at t = a and t = b, we have

$$ \begin{aligned} & \int\limits_{I} {\eta^{T} \left\{ {\sum\limits_{i = 1}^{p} {\lambda^{i} \left( {f_{u}^{i} \left( {t,\bar{u}\left( t \right),\dot{\bar{u}}\left( t \right)} \right) + B^{i} \left( t \right)w^{i} \left( t \right) - Df_{{\dot{u}}}^{i} \left( {t,\bar{u}\left( t \right),\dot{\bar{u}}\left( t \right)} \right)} \right.} } \right.} \\ & \quad + \left. {y\left( t \right)^{T} g_{u} \left( {t,\bar{u}\left( t \right),\dot{\bar{u}}\left( t \right)} \right) + z\left( t \right)^{T} h_{u} \left( {t,\bar{u}\left( t \right),\dot{\bar{u}}\left( t \right)} \right)} \right) \\ & \quad - \left. { D\left( {f_{{\dot{u}}}^{i} + y\left( t \right)^{T} g_{{\dot{u}}} + z\left( t \right)^{T} h_{{\dot{u}}} \left( {t,u,\dot{u}} \right)} \right) + H\beta \left( t \right)} \right\}dt < 0. \\ \end{aligned}$$

contradicts the equality constraint of the dual variational problem (NWVED). Hence

$$ \bar{x}\left( t \right) = \bar{u}\left( t \right), t \in I. $$

The following is the Huard [6] type converse duality:

Theorem 6 (Converse duality):

Let $\left( {\bar{x},\lambda ,y,z,w^{1} , \ldots ,w^{p} ,\beta \left( t \right)} \right)$ be an efficient solution of (NWVED) for which

$$\begin{array}{*{20}l} {\left( {{\text{C}}_{{\text{1}}} } \right)} & {{\text{H}}\,{\text{ is}}\,{\text{non-singular}}} \\ {\left( {{\text{C}}_{{\text{2}}} } \right)} & {\left[ \begin{aligned} & \left({\sigma \left( t \right)^{T} H^{i} \sigma \left( t \right)} \right)_{x} - D\left( {\sigma \left( t \right)^{T} H^{i} \sigma \left( t \right)} \right)_{{\mathop x\limits^{.}}} + D^{2} \left( {\sigma \left( t \right)^{T} H^{i} \sigma \left( t \right)} \right)_{{\mathop x\limits^{..}}} \\ & - D^{3} \left( {\sigma \left( t \right)^{T} H^{i} \sigma \left( t \right)} \right)_{{\mathop x\limits^{...}}} + D^{4} \left( {\sigma \left( t \right)^{T} H^{i} \sigma \left( t \right)} \right)_{{\mathop x\limits^{....}}} \\ \end{aligned} \right]} \\ &\quad { + 2\left[ \begin{aligned} & \sigma \left( t \right)^{T} \left( {H\sigma \left( t \right)} \right)_{x} - \sigma \left( t \right)^{T} D\left( {H\sigma \left( t \right)} \right)_{{\mathop x\limits^{.}}} + \sigma \left( t \right)^{T} D^{2} \left( {H\sigma \left( t \right)} \right)_{{\mathop x\limits^{..}}} \\ & - \sigma \left( t \right)^{T} D^{3} \left( {H\sigma \left( t \right)} \right)_{{\mathop x\limits^{...}}} + \sigma \left( t \right)^{T} D^{4} \left( {H\sigma \left( t \right)} \right)_{{\ddddot {x}}} \\ \end{aligned} \right] = 0 \Rightarrow \sigma \left( t \right) = 0,} \\ \end{array} $$

where $\sigma \left( t \right)$ is a vector function.

Then $\bar{x}\left( t \right)$ is feasible for (NWVEP) and the two objectives functional have the same value. Also, if the weak duality theorem holds for all feasible of (NWVEP) and (NWVED), then $\bar{x}\left( t \right)$ is efficient.

Proof:

Since $\left( {\bar{x},\lambda ,y,z,w^{1} , \ldots ,w^{p} ,\beta \left( t \right)} \right)$ is an efficient solution of (NWVED), there exists $\alpha ,\xi \in R^{p} ,\delta \in R$ and piecewise smooth $\theta :\,I \to R^{n} ,$ $\eta :\,I \to R^{m} ,$ $q^{i} :\,I \to R, i \in K$ such that following Fritz John conditions (Theorem 1)are satisfied at $\left( {\bar{x},\lambda ,y,z,w^{1} , \ldots ,w^{p} ,\beta \left( t \right)} \right) :$

$$ \begin{aligned} & \sum\limits_{{i = 1}}^{p} {\alpha ^{i} } \left[ {\left( {f_{x}^{i} + B^{i} \left( t \right)w^{i} \left( t \right) + y^{T} g_{x} + z^{T} h_{x} } \right) - D\left( {f_{{\mathop x\limits^{.}}}^{i} + y^{T} g_{{\mathop x\limits^{.}}} + z^{T} h_{{\mathop x\limits^{.}}} } \right)} \right. \\ & \quad - \frac{1}{2}\left( {\beta \left( t \right)^{T} H^{i} \beta \left( t \right)} \right)_{x} + \frac{1}{2}D\left( {\beta \left( t \right)^{T} H^{i} \beta \left( t \right)} \right)_{{\mathop x\limits^{.}}} - \frac{1}{2}D^{2} \left( {\beta \left( t \right)^{T} H^{i} \beta \left( t \right)} \right)_{{\mathop x\limits^{..}}} \\ & \quad + \left. { \frac{1}{2}D^{3} \left( {\beta \left( t \right)^{T} H^{i} \beta \left( t \right)} \right)_{{\mathop x\limits^{...}}} - \frac{1}{2}D^{4} \left( {\beta \left( t \right)^{T} H^{i} \beta \left( t \right)} \right)_{{\mathop x\limits^{....}}} } \right] \\ & \quad + \theta \left( t \right)^{T} \left[ {H + \left( {H\beta } \right)_{x} - D\left( {H\beta } \right)_{{\mathop x\limits^{.}}} + D^{2} \left( {H\beta } \right)_{{\mathop x\limits^{..}}} - D^{3} \left( {H\beta } \right)_{{\mathop x\limits^{...}}} + D^{4} \left( {H\beta } \right)_{{\mathop x\limits^{....}}} } \right] = 0 \\ \end{aligned}$$

(10)

$$ \begin{aligned} & \left( {\alpha ^{T} e} \right)\left[ {g^{j} - \frac{1}{2}\beta \left( t \right)^{T} g_{{xx}}^{j} \beta \left( t \right)} \right] + \theta \left( t \right)^{T} \left[ {g_{x}^{j} - Dg_{{\mathop x\limits^{.}}}^{j} } \right. \\ & \quad + \left. { \left( {g_{{xx}}^{j} - 2Dg_{{x\dot{x}}}^{j} + D^{2} g_{{\dot{x}\dot{x}}}^{j} - D^{3} g_{{\dot{x}\mathop {\text{x}}\limits^{{..}} }}^{j} } \right)\beta \left( t \right)} \right] + \eta ^{i} \left( t \right) = 0 \\ \end{aligned} $$

(11)

$$ \begin{aligned} & \left( {\alpha ^{T} e} \right)\left[ {h - \frac{1}{2}\beta \left( t \right)^{T} h_{{xx}} \beta \left( t \right)} \right]\; + \theta \left( t \right)^{T} \left[ {h_{x} - Dh_{{\mathop x\limits^{.}}} } \right. \\ & \quad + \left. { \left( {h_{{xx}} - 2Dh_{{x\dot{x}}} + D^{2} h_{{\dot{x}\dot{x}}} - D^{3} h_{{\dot{x}\mathop x\limits^{{..}} }} } \right)\beta \left( t \right)} \right] = 0 \\ \end{aligned} $$

(12)

$$ \left( {\theta \left( t \right) - \left( {\alpha^{T} e} \right)\beta } \right)H = 0 $$

(13)

$$ \theta \left( t \right)^{T} \left( {f_{x}^{i} + B^{i} \left( t \right) - Df_{{\mathop x\limits^{.}}}^{i} - H^{i} \beta (t)} \right) + \xi^{i} + \delta^{i} = 0 $$

(14)

$$ \alpha^{i} B^{i} \left( t \right)\bar{x}\left( t \right) + \theta B^{i} \left( t \right) - 2q^{i} \left( t \right)B^{i} \left( t \right)w^{i} \left( t \right) = 0,\quad t \in I,i \in K $$

(15)

$$ q^{i} \left( t \right)\left( {1 - w^{i} \left( t \right)^{T} B^{i} \left( t \right)w^{i} \left( t \right)} \right) = 0 $$

(16)

$$ \eta \left( t \right)^{T} y\left( t \right) = 0,\quad t \in I $$

(17)

$$ \xi^{i} \lambda = 0 $$

(18)

$$ \delta \left( {\sum\limits_{i = 1}^{p} {\lambda^{i} - 1} } \right) = 0 $$

(19)

$$ \left( {\alpha ,\eta ,\xi ,q\left( t \right),\delta } \right)\geqq 0 $$

(20)

$$ \left( {\alpha ,\theta \left( t \right),\eta ,\xi ,q\left( t \right)\delta } \right) \ne 0 $$

(21)

Since $\lambda > 0,$ (18) implies $\xi = 0.$

Since H is non singular, (13) implies

$$ \theta \left( t \right) = \left( {\alpha^{T} e} \right)\beta \left( t \right),\quad t \in I $$

(22)

Using the equality constraint of the dual and (22), we

$$\begin{array}{*{20}l} { - \sum {\alpha ^{i} } 0\left[ {\begin{array}{*{20}l} {H\beta \left( t \right)^{T} - \frac{1}{2}\left( {\beta \left( t \right)^{T} H^{i} \beta \left( t \right)} \right)_{x} + \frac{1}{2}D\left( {\beta \left( t \right)^{T} H^{i} \beta \left( t \right)} \right)_{{\dot{x}}} - \frac{1}{2}D^{2} \left( {\beta \left( t \right)^{T} H^{i} \beta \left( t \right)} \right)_{{\ddot{x}}} } \\ \quad + { \frac{1}{2}D^{3} \left( {\beta \left( t \right)^{T} H^{i} \beta \left( t \right)} \right)_{{\dddot x}} - \frac{1}{2}D^{4} \left( {\beta \left( t \right)^{T} H^{i} \beta \left( t \right)} \right)_{{\ddddot x}} } \\ \end{array} } \right]} \\ {\quad + \left( {\alpha ^{T} e} \right)\left[ {\begin{array}{*{20}l} {H\beta \left( t \right)^{T} + \beta \left( t \right)^{T} \left( {H\beta } \right)_{x} - \beta \left( t \right)^{T} D\left( {H\beta } \right)_{{\dot{x}}} + \beta \left( t \right)^{T} D^{2} \left( {H\beta } \right)_{{\ddot{x}}} } \\ {\quad - \beta \left( t \right)^{T} D^{3} \left( {H\beta } \right)_{{\dddot x}} + \beta \left( t \right)^{T} D^{4} \left( {H\beta } \right)_{{\ddddot x}} } \\ \end{array} } \right] = 0} \\ \quad {\left[ {\begin{array}{*{20}l} {\left( {\beta \left( t \right)^{T} H^{i} \beta \left( t \right)} \right)_{x} - D\left( {\beta \left( t \right)^{T} H^{i} \beta \left( t \right)} \right)_{{\dot{x}}} + D^{2} \left( {\beta \left( t \right)^{T} H^{i} \beta \left( t \right)} \right)_{{\ddot{x}}} } \\ {\quad - D^{3} \left( {\beta \left( t \right)^{T} H^{i} \beta \left( t \right)} \right)_{{\dddot x}} + D^{4} \left( {\beta \left( t \right)^{T} H^{i} \beta \left( t \right)} \right)_{{\ddddot x}} } \\ \end{array} } \right]} \\\quad + 2\left[ \begin{gathered} \beta \left( t \right)^{T} \left( {H\beta } \right)_{x} - \beta \left( t \right)^{T} D\left( {H\beta } \right)_{{\dot{x}}} + \beta \left( t \right)^{T} D^{2} \left( {H\beta } \right)_{{\ddot{x}}} \hfill \\ - \beta \left( t \right)^{T} D^{3} \left( {H\beta } \right)_{{\dddot x}} + \beta \left( t \right)^{T} D^{4} \left( {H\beta } \right)_{{\ddddot x}} \hfill \\ \end{gathered} \right] = 0 \end{array} $$

This because of the hypothesis (C₂), yields $\beta \left( t \right) = 0,t \in I$

Using $\beta \left( t \right) = 0,t \in I$ in (22), we have $\theta \left( t \right) = 0, t \in I$

Let $\alpha^{i} = 0, i \in K,$ then (11) and (14) respectively give $\eta^{i} = 0$ and $\delta^{i} = 0, i \in K.$

Consequently (15) and (16) imply $q^{i} \left( t \right) = 0, t \in I, i \in K$

Thus $\left( {\alpha ,\theta \left( t \right),\eta ,\xi ,q\left( t \right),\delta } \right) = 0, i \in K,$ where $q\left( t \right) = \left( {q^{1} \left( t \right),q^{2} \left( t \right), \ldots ,q^{p} \left( t \right)} \right),$ contradicting the Fritz John condition (21). Hence $\alpha^{i} > 0, i \in K.$

Using $\theta \left( t \right) = 0,\; \alpha > 0$ and $\beta \left( t \right) = 0,\;t \in I$, from (11), we have

$$ g^{j} = - \frac{{\eta^{i} \left( t \right)}}{{\left( {\alpha^{T} e} \right)}}\leqq 0,\quad t \in I, $$

(23)

yielding

$$ g\left( {t,x,\dot{x}} \right)\leqq 0,\quad t \in I $$

The relations (23) and (12) respectively imply that

$$ y\left( t \right)^{T} g\left( {t,x,\dot{x}} \right) = 0,\quad t \in I\;\text{and}\;z\left( t \right)^{T} h\left( {t,x,\dot{x}} \right) = 0 $$

(24)

Also $g\left( {t,x,\dot{x}} \right)\leqq 0$ and $h\left( {t,x,\dot{x}} \right) = 0,\;t \in I$ imply that $\bar{x}$ is feasible for (NWVEP).

Using $\theta \left( t \right) = 0,\;t \in I,\alpha^{i} > 0,\;i \in K,$ (15) implies

$$ B^{i} \left( t \right)\bar{x}\left( t \right) = \left( {\frac{{2q^{i} \left( t \right)}}{{\alpha^{i} }}} \right)B^{i} \left( t \right)w^{i} \left( t \right) = 0,\quad t \in I,\;i \in K $$

(25)

This yields the equality in the Schwartz inequality. That is,

$$ \bar{x}\left( t \right)^{T} B\left( t \right)w^{i} \left( t \right) = \left( {\bar{x}\left( t \right)^{T} B\left( t \right)\bar{x}\left( t \right)} \right)^{1/2} \left( {w^{i} \left( t \right)^{T} B\left( t \right)w^{i} \left( t \right)} \right)^{1/2} ,\quad t \in I,\;i \in K $$

(26)

If $q^{i} \left( t \right) > 0,\;\forall i \in K,\;t \in I,$ then (16) gives $w^{i} \left( t \right)^{T} B\left( t \right)w^{i} \left( t \right) = 1, t \in I$ and so (26) gives $\bar{x}\left( t \right)^{T} B\left( t \right)w^{i} \left( t \right) = \left( {\bar{x}\left( t \right)^{T} B\left( t \right)\bar{x}\left( t \right)} \right)^{1/2},\;t \in I,\;i \in K$

If $q^{i} \left( t \right) = 0,\;\forall i \in K,\;t \in I ,$ then (15) gives $B^{i} \left( t \right)\bar{x}\left( t \right) = 0,\;\forall i \in K,t \in I$

So we will still have

$$ \bar{x}\left( t \right)^{T} B\left( t \right)w^{i} \left( t \right) = \left( {\bar{x}\left( t \right)^{T} B\left( t \right)\bar{x}\left( t \right)} \right)^{1/2} ,\quad t \in I,i \in K $$

(27)

In view of (24) and (26) we have

$$ \begin{aligned} & \int\limits_{I} {\left\{ {f^{i} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right) + \left( {\bar{x}\left( t \right)^{T} B^{i} \left( t \right)\bar{x}\left( t \right)} \right)^{1/2}} \right\}dt} = \int\limits_{I} {\left( {f^{i} \left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right) + \bar{x}\left( t \right)^{T} B^{i} \left( t \right)w^{i} \left( t \right)} \right.} \\ & \quad +\left. {y\left( t \right)^{T} g\left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right) + z\left( t \right)^{T} h\left( {t,\bar{x}\left( t \right),\dot{\bar{x}}\left( t \right)} \right) - \frac{1}{2}\beta \left( t \right)^{T} H^{i} \beta \left( t \right)} \right)dt \\ \end{aligned} $$

i.e. the objective values of the problem are equal. By Theorem 1 the efficiency of $\bar{x}\left( t \right)$ for (NWVEP) follows.

5 Nondifferentiable Multiobjective Variational Problems with Natural Boundary Values

The following is a pair of Wolfe type second-order nondifferentiable multiobjective variational problems with natural boundary values:

$\begin{aligned} {\left( {{\mathbf{NVEP}}} \right)_{{\text{1}}} {\text{:}}} \quad & {{\text{Minimize}}} \\ & {\begin{array}{*{20}l} {\begin{array}{*{20}l} {\left( {\int\limits_{I} {\left( {f^{1} \left( {t,x\left( t \right),\dot{x}\left( t \right)} \right) + \left( {x\left( t \right)^{T} B^{1} \left( t \right)x\left( t \right)} \right)^{1/2}} \right)} dt,\ldots,} \right.} \\ {\left. {\int\limits_{I} {\left( {f^{p} \left( {t,x\left( t \right),\dot{x}\left( t \right)} \right) + \left( {x\left( t \right)^{T} B^{p} \left( t \right)x\left( t \right)} \right)^{1/2} } \right)} dt} \right)} \\ \end{array} } \\ \end{array}} \\ \quad & {\text{subject}\,\text{to}} \quad {g\left( {t,x,\dot{x}} \right){ \leqq }0,h\left( {t,x,\dot{x}} \right) = 0,\quad t \in I} \\ \end{aligned}$

$\begin{array}{*{20}l} {\left( {{\mathbf{WNVED}}} \right)_{{\text{2}}} {\text{:}}} & {{\text{Maximize}}} \\ {} & {\left[ {\int\limits_{I} {\left( {f^{1} \left( {t,u\left( t \right),\dot{u}\left( t \right)} \right) + u\left( t \right)^{T} B^{1} \left( t \right)w^{1} \left( t \right)} \right.} } \right.} \\ {} & {\left. {\quad + y\left( t \right)^{T} g\left( {t,u\left( t \right),\dot{u}\left( t \right)} \right) + z\left( t \right)^{T} h\left( {t,u\left( t \right),\dot{u}\left( t \right)} \right)} \right)dt,} \\ {} & {\ldots,\int\limits_{I} {\left( {f^{p} \left( {t,u\left( t \right),\dot{u}\left( t \right)} \right) + u\left( t \right)^{T} B^{p} \left( t \right)w^{p} \left( t \right)} \right.} } \\ {} & {\left. {\left. {\quad + y\left( t \right)^{T} g\left( {t,u\left( t \right),\dot{u}\left( t \right)} \right) + z\left( t \right)^{T} h\left( {t,u\left( t \right),\dot{u}\left( t \right)} \right)} \right)dt} \right]} \\ {} & {\text{subject to}} \\ {} & \quad {\begin{aligned} & \sum\limits_{i = 1}^{p} {\lambda^{i} } \left( {f_{u}^{i} \left( {t,u\left( t \right),\dot{u}\left( t \right)} \right) + B^{i} \left( t \right)w^{i} \left( t \right) - D\,f_{{\dot{u}}}^{i} \left( {t,u\left( t \right),\dot{u}\left( t \right)} \right) + y\left( t \right)^{T} g_{u} \left( {t,u\left( t \right),\dot{u}\left( t \right)} \right)} \right. \\ & \left. { + z\left( t \right)^{T} h_{u} \left( {t,u\left( t \right),\dot{u}\left( t \right)} \right) - D\left( {y\left( t \right)^{T} g_{{\dot{u}}} \left( {t,u\left( t \right),\dot{u}\left( t \right)} \right) + z\left( t \right)^{T} h_{{\dot{u}}} \left( {t,u\left( t \right),\dot{u}\left( t \right)} \right)} \right)} \right) = 0,\quad t \in I \\ & w^{i} \left( t \right)^{T} B^{i} \left( t \right)w^{i} \left( t \right)\leqq 1,\quad i \in K \\ & y\left( t \right)\geqq 0,t \in I,\lambda > 0,\lambda^{T} e = 1. \\ & \lambda^{T} f_{{\dot{u}}} \left( {t,u\left( t \right),\dot{u}\left( t \right)} \right) + y\left( t \right)^{T} g_{{\dot{u}}} \left( {t,u\left( t \right),\dot{u}\left( t \right)} \right) + z\left( t \right)^{T} h_{{\dot{u}}} \left( {t,u\left( t \right),\dot{u}\left( t \right)} \right) = 0,\quad t \in I \\ \end{aligned}} \end{array}$

The duality theorems validated in the preceding section can easily be established with slight modifications.

6 Wolfe Type Second-Order Nondifferentiable Multiobjective Mathematical Programming Problems

If f, g and h variational problems (NWVEP) and (NWVED) are independent of t i.e. functions do not depend explicitly on t, these problems reduce to the following second-order nondifferentiable multiobjective mathematical programming problems:

$$ \begin{array}{*{20}l} {\left( {{\mathbf{NVEP}}} \right)_{{\mathbf{0}}} {\mathbf{:}}} & {{\text{Minimize}}} & {\left( {f^{1} \left( x \right) + \left( {x^{T} B^{1} x} \right)^{{1/2}} ,\ldots,f^{p} \left( x \right) + \left( {x^{T} B^{p} x} \right)^{{1/2}}} \right)} \\ {} & {{\text{subject to}}} & {g(x)\leqq0\quad h\left( x \right) = 0} \\ \end{array} $$

$$ \begin{aligned} {{\mathbf{NWVED}}}:\;&{\text{Maximize}} \\ & \left( {f^{1} \left( u \right) + u^{T} B^{1} w^{1} + y^{T} g\left( u \right) + z^{T} h\left( u \right) - \frac{1}{2}\beta^{T} \nabla^{2} \left( {\lambda {}^{T}f^{1} \left( u \right) + y^{T} g\left( u \right) + z^{T} h\left( u \right)} \right)\beta , \ldots ,} \right. \\ & \left. {f^{p} \left( u \right) + u^{T} B^{p} w^{p} + y^{T} g\left( u \right) + z^{T} h\left( u \right) - \frac{1}{2}\beta^{T} \nabla^{2} \left( {\lambda^{T} f^{p} \left( u \right) + y^{T} g\left( u \right) + z^{T} h\left( u \right)} \right)\beta } \right) \\ & {\begin{array}{*{20}l} {{\text{subject to}}} & {} & {\sum {\lambda ^{T} } \left( {f_{u}^{i} \left( u \right) + B^{i} w^{i} } \right) + y^{T} g_{u} \left( u \right) + z^{T} h_{u} \left( u \right) = 0} \\ {} & {} & {w^{iT} B^{i} w^{i} \,\leqq\,1} \\ {} & {} & {y\geqq0,\,\,\lambda > 0,\lambda ^{T} e = 1,e = \left( {1,\ldots,1} \right).} \\ \end{array}}\end{aligned} $$

7 Conclusion

Both Fritz John and Karush-Kuhn-Tucker type optimality conditions for a class of nondifferentiable multiobjective variational problems with equality and inequality constraints are obtained. Here the nondifferentiability occurs due to appearance of a square root function in each component of the objective functional. As an application of Karush-Kuhn-Tucker type optimality conditions Wolfe and Mond-Weir type duals to the problem treated in the research are formulated. Lastly, the relationship between our results and those of their static counterparts has been indicated. The research exposition in this paper has scope to revisit in the context of mixed type second-order duality which has some computational advantage over either of Wolfe or Mond-Weir type duality.