Robust Duality for Multi-dimensional Variational Control Problem with Data Uncertainty

Abstract

Duality is a powerful tool to study optimization problems. This is because duality principles relate a pair of optimization problems, known as primal and dual problems, in such a way that the existence of solution of one ensures the solution of the other and the extrema of two problems coincide under certain conditions. The duality theorems for linear optimization were conjectured by Neumann [1]. Thereafter, Wolfe [2] introduced a dual model known as “Wolfe dual” for a class of nonlinear convex optimization problems and established the duality results. Later, Mond and Weir [3] introduced the “Mond-Weir dual” whose objective function is simple as compared to the objective function of the Wolfe dual. After that, many researchers obtained duality results for various optimization problems under different assumptions. Mond and Smart [4] studied the control problem and derived the duality results under the invexity assumption. Thereafter, Mititelu [5] formulated Wolfe’s dual for the multi-time control problem and established the duality results under the invexity hypotheses. Beck and Ben-Tal [6] investigated the solution of the uncertain optimization problem via duality and shown that the primal worst case is equal to the dual best case. Jeyakumar et al. [7] also studied the robust duality for uncertain optimization problem under generalized convexity.

7.1 Introduction

Duality is a powerful tool to study optimization problems. This is because duality principles relate a pair of optimization problems, known as primal and dual problems, in such a way that the existence of solution of one ensures the solution of the other and the extrema of two problems coincide under certain conditions. The duality theorems for linear optimization were conjectured by Neumann [1]. Thereafter, Wolfe [2] introduced a dual model known as “Wolfe dual” for a class of nonlinear convex optimization problems and established the duality results. Later, Mond and Weir [3] introduced the “Mond-Weir dual” whose objective function is simple as compared to the objective function of the Wolfe dual. After that, many researchers obtained duality results for various optimization problems under different assumptions. Mond and Smart [4] studied the control problem and derived the duality results under the invexity assumption. Thereafter, Mititelu [5] formulated Wolfe’s dual for the multi-time control problem and established the duality results under the invexity hypotheses. Beck and Ben-Tal [6] investigated the solution of the uncertain optimization problem via duality and shown that the primal worst case is equal to the dual best case. Jeyakumar et al. [7] also studied the robust duality for uncertain optimization problem under generalized convexity. Recently, Treanţă and Mititelu [8], extended the duality theory over multi-dimensional fractional control problem under the assumption of $(\rho ,b)$ quasiinvexity.

7.2 Problem Description

Throughout the chapter, we are considering the following notations and working hypotheses:

$\bullet$ $\mathbb {R}^p, \mathbb {R}^q, \mathbb {R}^m$ and $\mathbb {R}^n$ are Euclidean spaces of dimensions p, q, r and n, respectively.

$\bullet$ $\varGamma _{t_{0},t_{1}} \subset \mathbb {R}^p$ is a hyperparallelepiped fixed by the diagonally opposite points $t_{0}=(t_{0}^\alpha )$ and $t_{1}=(t_{1}^\alpha ),\alpha =\overline{1,p}$ and the point $t=((t^{\alpha }),\alpha =\overline{1,p}) \in \varGamma _{t_{0},t_{1}} \subset \mathbb {R}^p$.

$\bullet$ $\mathcal {X}$ is the space of state functions (piecewise smooth) $x=(x^{\tau }):\varGamma _{t_{0},t_{1}} \subset \mathbb {R}^p \rightarrow \mathbb {R}^q$ and $\frac{\partial x}{\partial t^\alpha }=x_{\alpha }$ denotes the partial derivative of x with respect to $t^\alpha$.

$\bullet$ $\mathscr {C}$ is the space of control functions (piecewise continuous) $c=(c^{j}): \varGamma _{t_{0},t_{1}} \subset \mathbb {R}^p \rightarrow \mathbb {R}^m$.

$\bullet$ $dt=dt^1 \dots dt^p$ is the volume element on $\mathbb {R}^p\supset \varGamma _{t_{0},t_{1}}$.

$\bullet$ T denotes the transpose of a vector.

$\bullet$ We assume the following convention for inequalities and equalities: for $x,y \in \mathbb {R}^n$, we have

(i):

$x< y \Leftrightarrow x_{i}<y_{i},\forall i=\overline{1, n},$

(ii):

$x= y\Leftrightarrow x_{i}=y_{i}, ~\forall i=\overline{1, n},$

(iii):

$x\leqq y \Leftrightarrow x_{i}\le y_{i}, ~\forall i=\overline{1, n},$

(iv):

$x\le y \Leftrightarrow x_{i}\le y_{i}, ~\forall i=\overline{1, n} ~\text{ and }~ x_{i}<y_{i} ~\text{ for } \text{ some }~i.$

The multi-dimensional multi-objective optimization problem with data uncertainty in the objective and constraint functionals is defined as follows:

$$\begin{aligned} \text {(MMOPU)}~~~&\min _{(x(\cdot ),c(\cdot ))} \Big \{\int _{\varGamma _{t_{0},t_{1}}} \mathcal {f}(t,x(t),c(t),w)dt\\&=\Big ( \int _{\varGamma _{t_{0},t_{1}}} \mathcal {f}_{1}(t,x(t),c(t),w_{1})dt, \dots , \int _{\varGamma _{t_{0},t_{1}}} \mathcal {f}_{s}(t,x(t),c(t),w_{s})dt \Big )\Big \} \\ ~~~ \text {subject to}~&~~~G(t,x(t),c(t), u)\leqq 0, \\&~~~ x_{\alpha }= H (t,x(t),c(t), v), \\&~~~x(t_{0})=x_{0}, ~x(t_{1})=x_{1}, \end{aligned}$$

where $\mathcal {f}=(\mathcal {f}_{1},\dots ,\mathcal {f}_{s} );(\mathcal {f}_{k}): \varGamma _{t_{0},t_{1}} \times \mathscr {X} \times \mathscr {C} \times \mathcal {W}_{k} \rightarrow \mathbb {R}^s, k=\overline{1,s}$, $G=(G_{1},\dots , G_{m});(G_{l}):\varGamma _{t_{0},t_{1}} \times \mathscr {X} \times \mathscr {C} \times \mathcal {U}_{l} \rightarrow \mathbb {R}^m,~l=\overline{1,m}$, $H=(H^{\tau }_{\alpha }): \varGamma _{t_{0},t_{1}} \times \mathscr {X} \times \mathscr {C} \times \mathcal {V}^\tau _ {\alpha } \rightarrow \mathbb {R}^{pq}, \alpha =\overline{1,p}, \tau =\overline{1,q}$ are $C^{\infty }$-class functionals. Consider $w=(w_{k}), u=(u_{l})$ and $v=(v^\tau _{\alpha })$ are the uncertain parameters for some convex compact subsets $\mathcal {W}=(\mathcal {W}_{k}) \subset \mathbb {R}^s, \mathcal {U}=(\mathcal {U}_{l})\subset \mathbb {R}^m$ and $\mathcal {V}=(\mathcal {V}^\tau _ {\alpha }) \subset \mathbb {R}^{pq}$, respectively. The functions $H_{\alpha }$ satisfy the closeness conditions (complete integrability conditions) $D_{\beta }H_{\alpha }=D_{\alpha }H_{\beta },\alpha ,\beta =\overline{1,p}, \alpha \not = \beta ,$ where the total derivative is denoted by $D_{\beta }$.

The associated robust counterpart of the multi-dimensional multi-objective optimization problem (MMOPU) is defined as:

$$\begin{aligned} \text {(RMMOPU)}~&~~\min _{(x(\cdot ),c(\cdot ))}\Big \{\int _{\varGamma _{t_{0},t_{1}}} \max _{w\in \mathcal {W}}\mathcal {f}(t,x(t),c(t),w)dt\\&=\Big ( \int _{\varGamma _{t_{0},t_{1}}} \max _{w_{1}\in \mathcal {W}_{1}}\mathcal {f}_{1}(t,x(t),c(t),w_{1})dt, \dots , \int _{\varGamma _{t_{0},t_{1}}} \max _{w_{s}\in \mathcal {W}_{s}}\mathcal {f}_{s}(t,x(t),c(t),w_{s})dt \Big )\Big \} \\ ~~~ \text {subject to}~&~~~~~~~~~G(t,x(t),c(t), u)\leqq 0, \forall u\in \mathcal {U}, \\&~~~~~~~~~x_{\alpha }= H (t,x(t),c(t), v), \forall v\in \mathcal {V}, \\&~~~~~~~~~ x(t_{0})= x_{0}, ~x(t_{1})=x_{1}, \end{aligned}$$

where $\mathcal {f},G~\text {and}~H$ are defined as above in the multi-dimensional multi-objective optimization problem (MMOPU).

We denote $\mathcal {D}=\lbrace (x, c) \in \mathscr {X}\times \mathscr {C} :~G(t,x(t),c(t), u)\leqq 0, x_{\alpha }= H (t,x(t),c(t), v), x(t_{0})=x_{0},x(t_{1})=x_{1} \rbrace$ the set of all feasible solutions to (RMMOPU) and we say that it is the robust feasible solution set to the problem (MMOPU).

From now on, to simplify our presentation, we introduce some notations, as follows: $x=x(t), \hat{x}=\hat{x}(t), \bar{x}=\bar{x}(t), \tilde{x}=\tilde{x}(t), c=c(t), \hat{c}=\hat{c}(t), \bar{c}=\bar{c}(t), \tilde{c}=\tilde{c}(t), \pi =(t,x(t),c(t)), \bar{\pi }=(t,\bar{x}(t),\bar{c}(t)), \hat{\pi }=(t,\hat{x}(t),\hat{c}(t)), \tilde{\pi }=(t, \tilde{x}(t), \tilde{c}(t)), \eta =(t,y(t), z(t)), \bar{\eta }=(t, \bar{y}(t), \bar{z}(t)), \rho =\rho (t), \bar{\rho }=\bar{\rho }(t)$ and $\varGamma =\varGamma _{t_{0},t_{1}}$.

The partial derivatives of Lagrange multiplier $\gamma$ and the partial derivatives associated with $\mathcal {f}$ are defined as

$$\begin{aligned} \mathcal {\gamma }_{t} = \begin{bmatrix} \frac{\partial \gamma ^{\tau }_{1}}{\partial t^{1}} &{} \cdots &{} \frac{\partial \gamma ^{\tau }_{1}}{\partial t^{p}} \\ \vdots &{} \ddots &{} \vdots \\ \frac{\partial \gamma ^{\tau }_{p}}{\partial t^{1}} &{} \cdots &{} \frac{\partial \gamma ^{\tau }_{p}}{\partial t^{p}} \end{bmatrix}, \quad \mathcal {f}_{x} = \begin{bmatrix} \frac{\partial \mathcal {f}_{1}}{\partial x^{1}} &{} \cdots &{} \frac{\partial \mathcal {f}_{1}}{\partial x^{q}} \\ \vdots &{} \ddots &{} \vdots \\ \frac{\partial \mathcal {f}_{s}}{\partial x^{1}} &{} \cdots &{} \frac{\partial \mathcal {f}_{s}}{\partial x^{q}} \end{bmatrix}, \quad \mathcal {f}_{c} = \begin{bmatrix} \frac{\partial \mathcal {f}_{1}}{\partial c^{1}} &{} \cdots &{} \frac{\partial \mathcal {f}_{1}}{\partial c^{r}} \\ \vdots &{} \ddots &{} \vdots \\ \frac{\partial \mathcal {f}_{s}}{\partial c^{1}} &{} \cdots &{} \frac{\partial \mathcal {f}_{s}}{\partial c^{r}} \end{bmatrix}. \end{aligned}$$

Similarly, we have $G_{x}$ and $G_{c}$ using matrices with m rows and $H_{x}, H_{c}, x_{\alpha }$ using matrices with p rows.

In general, for a vector optimization problem, it is not so easy to find a point which optimizes more than one objective simultaneously. Therefore, the notion of an efficient solution is used to determine an optimality in vector optimization problems.

Definition 7.2.0.1

A point $(\bar{x},\bar{c})\in \mathcal {D}$ is said to be weak robust efficient solution to the multi-dimensional multi-objective optimization problem (MMOPU), if there does not exist another point $(x,c)\in \mathcal {D}$ such that

$$\begin{aligned} \int _{\varGamma }\max _{w\in \mathcal {W}}\mathcal {f}( \pi , w) dt < \int _{\varGamma }\max _{w\in \mathcal {W}}\mathcal {f}(\bar{\pi }, w)dt. \end{aligned}$$

Definition 7.2.0.2

A functional $F(x, c,\bar{w})=\int _{\varGamma }\mathcal {f}(t, x(t), x_{\sigma }(t), c(t), \bar{w}) dt$ is said to be convex (strict convex) at $(\bar{x}, \bar{c})$, if the following inequality

$$\begin{aligned} F(x, c,\bar{w})-F(\bar{x}, \bar{c},\bar{w}) \geqq (>)&\int _{\varGamma } (x(t)-\bar{x}(t)) \mathcal {f}_{x} (t,\bar{x}(t),\bar{x}_{\sigma }(t),\bar{c}(t),\bar{w})dt \\&~~+ \int _{\varGamma } (x_{\sigma }(t)-\bar{x}_{\sigma }(t)) \mathcal {f}_{x_{\sigma }} (t,\bar{x}(t),\bar{x}_{\sigma }(t),\bar{c}(t),\bar{w})dt \\&~~ +\int _{\varGamma } (c(t)-\bar{c}(t)) \mathcal {f}_{c} (t, \bar{x}(t),\bar{x}_{\sigma }(t),\bar{c}(t),\bar{w})dt \end{aligned}$$

holds for all (x(t), c(t)) on $\mathscr {X} \times \mathscr {C}$.

Following the same lines formulated in Mititelu [5], we establish the following result. Thus, the proof is omitted.” The proof can be read (it is free) at the following link: https://www.journal.fairpartners.ro/volume-22009-no-1_3.html

Theorem 7.2.0.1

Let $(\bar{x},\bar{c})\in \mathcal {D}$ be a weak robust efficient solution to the problem $\text {(MMOPU)}$. Further assume that $\max _{w\in \mathcal {W}}\mathcal {f} (\pi , w)=\mathcal {f}(\pi , \bar{w})$. If the constraint conditions (for the existence of the multipliers) hold, then there exist the scalar vector $\bar{\mu }\in \mathbb {R}^s$, the piecewise smooth Lagrange multipliers $\bar{\nu }=(\bar{\nu }_{l}(t))\in \mathbb {R}^{m}_{+},l \in \mathbb {R}^m, \bar{\gamma }=(\bar{\gamma }^{\tau }_\alpha (t))\in \mathbb {R}^{pq},\tau =\overline{1,q},\alpha =\overline{1,p}$, and the uncertain parameters $\bar{u}\in \mathcal {U}, \bar{v}\in \mathcal {V}$ such that $(\bar{x}, \bar{c})$ satisfies the following conditions:

$$\begin{aligned}&\bar{\mu }^{T} \mathcal {f}_{x} (\bar{\pi }, \bar{w})+\bar{\nu }^{T}G_{x} (\bar{\pi }, \bar{u})+\bar{\gamma }^{T} H_{x} (\bar{\pi }, \bar{v})+\bar{\gamma }_{t}=0, \end{aligned}$$

(7.2.1)

$$\begin{aligned}&\bar{\mu }^{T} \mathcal {f}_{c} (\bar{\pi }, \bar{w})+\bar{\nu }^{T}G_{c} (\bar{\pi }, \bar{v})+\bar{\gamma }^{T} H_{c} (\bar{\pi }, \bar{v})=0, \end{aligned}$$

(7.2.2)

$$\begin{aligned}&\bar{\nu }^{T} G(\bar{\pi }, \bar{u})=0, \end{aligned}$$

(7.2.3)

$$\begin{aligned}&\bar{\mu }\ge 0,~\bar{\nu }\geqq 0, \end{aligned}$$

(7.2.4)

hold for all $t\in \varGamma ,$ except at discontinuities.

Remark 7.2.0.1

The conditions (7.2.1)–(7.2.4) are known as the robust necessary efficiency conditions for the multi-dimensional multi-objective optimization problem (MMOPU).

7.3 Wolfe Type Robust Duality

In this section, we formulate Wolfe type robust dual problem for the multi-dimensional multi-objective optimization problem with data uncertainty in the objective and constraint functionals (MMOPU) as follows:

$$\begin{aligned} \text {(UWRDP)}~~\max _{(y(\cdot ),z(\cdot ))}&\int _{\varGamma _{t_{0},t_{1}}} \Big \lbrace \mathcal {f}(t,y(t),z(t),w)+ \nu ^T G(t,y(t),z(t),u)e\nonumber \\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+\gamma ^T (H (t,y(t),z(t),v)-y_{\alpha })e \Big \rbrace dt \nonumber \\ ~~~ \text {subject to}~&~~~~ \mu ^{T}\mathcal {f}_{x} (t,y(t),z(t),w)+ \nu ^T G_{x} (t,y(t),z(t),u)\nonumber \\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~+ \gamma ^T H_{x} (t,y(t),z(t),v)+ \gamma _{t}=0, \end{aligned}$$

(7.3.1)

$$\begin{aligned}&~~~~\mu ^{T}\mathcal {f}_{c}(t,y(t),z(t),w)+ \nu ^T G_{c} (t,y(t),z(t),u)\nonumber \\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+ \gamma ^T H_{c}(t,y(t),z(t),v)=0, \end{aligned}$$

(7.3.2)

$$\begin{aligned}&~~~~y(t_{0})=x_{0}, y(t_{1})=x_{1}, \end{aligned}$$

(7.3.3)

$$\begin{aligned}&~~~~\mu ^{T}> 0,~ e^T \mu =1, e=(1, \dots , 1)\in \mathbb {R}^s. \end{aligned}$$

(7.3.4)

The associated robust counterpart for the problem (UWRDP) is given as:

$$\begin{aligned} \text {(RUWRDP)}~~~~\max _{(y(\cdot ),z(\cdot ), w,u,v)}&\int _{\varGamma _{t_{0},t_{1}}} \Big \lbrace \mathcal {f}(t,y(t),z(t),w)+ \nu ^T G(t,y(t),z(t),u)e\\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~+ \gamma ^T (H (t,y(t),z(t),v)-y_{\alpha })e \Big \rbrace dt \\ ~~~ \text {subject to}~&~~~~ \mu ^{T}\mathcal {f}_{x} (t,y(t),z(t),w)+ \nu ^T G_{x} (t,y(t),z(t),u)\\&~~~~~~~~~~~~~~~~~~+ \gamma ^T H_{x} (t,y(t),z(t),v)+ \gamma _{t}=0,\\&~~~~\mu ^{T}\mathcal {f}_{c}(t,y(t),z(t),w)+ \nu ^T G_{c} (t,y(t),z(t),u)\\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~+ \gamma ^T H_{c}(t,y(t),z(t),v)=0,\\&~~~~y(t_{0})=x_{0}, y(t_{1})=x_{1},\\&~~~~ \mu ^{T}> 0,~ e^T \mu =1, e=(1, \dots , 1)\in \mathbb {R}^s, \end{aligned}$$

for all $w \in \mathcal {W}, u\in \mathcal {U}, v\in \mathcal {V}.$

We denote $\mathcal {D_{w}}=\lbrace (y,z) \in \mathscr {X}\times \mathscr {C} :~\text{ satisfying } \text{ conditions }$ (7.3.1)–(7.3.4) $\rbrace$ the set of all feasible solutions to (RUWRDP) and we say that it is the robust feasible solution set to the problem (UWRDP).

Definition 7.3.0.1

A point $(\bar{y},\bar{z}, \bar{\mu }, \bar{\nu }, \bar{\gamma }, \bar{w}, \bar{u}, \bar{v} ) \in \mathcal {D_{w}}$ is said to be weak robust efficient solution to the Wolfe type robust dual problem (UWRDP), if there does not exist another point $(y,z, \mu , \nu , \gamma , w, u, v)\in \mathcal {D_{w}}$ such that

$$\begin{aligned} \int _{\varGamma _{t_{0},t_{1}}} \Big \lbrace \mathcal {f}(\bar{\eta }, \bar{w})+ \nu ^T G(\bar{\eta },\bar{u})&e+\gamma ^T (H (\bar{\eta }, \bar{v})-y_{\alpha })e \Big \rbrace dt \\&< \int _{\varGamma _{t_{0},t_{1}}} \Big \lbrace \mathcal {f}(\eta , \bar{w})+ \nu ^T G(\eta ,\bar{u})e +\gamma ^T (H (\eta ,\bar{v})-y_{\alpha })e \Big \rbrace dt. \end{aligned}$$

Now, we shall establish the weak duality result for (MMOPU) under convexity assumption which states that the value attained by the objective functional of the dual problem over its feasible set does not exceed the value attained by the objective functional of the primal problem.

Theorem 7.3.0.1

(Weak Robust Duality) Let $(\bar{x},\bar{c})$ and $( \bar{y},\bar{z},\bar{\mu },\bar{\nu },\bar{\gamma },\bar{w},\bar{u},\bar{v})$ be the robust feasible solutions of (MMOPU) and (UWRDP), respectively. Assume that $\max _{w\in \mathcal {W}} \mathcal {f}(\bar{\pi }, w) =\mathcal {f}(\bar{\pi }, \bar{w})$. Further, if $\int _{\varGamma } \bar{\mu }^T\mathcal {f}(\cdot ,\bar{w})dt, \int _{\varGamma } \bar{\nu }^T G(\cdot ,\bar{u}) dt,$ and $\int _{\varGamma } \bar{\gamma }^T \{ H (\cdot , \bar{v})- y_{\alpha } \} dt$ are convex at $(\bar{y},\bar{z})$, then the following inequality cannot hold

$$\begin{aligned} \int _{\varGamma } \max _{w\in \mathcal {W}} \mathcal {f}(\bar{\pi }, w) dt < \int _{\varGamma } \Big \lbrace \mathcal {f}(\bar{\eta }, \bar{w})+\bar{\nu }^T G(\bar{\eta },\bar{u})e+\bar{\gamma }^T ( H (\bar{\eta },\bar{v})-y_{\alpha }) e\Big \rbrace dt. \end{aligned}$$

Proof

Contrary to the result, we assume that

$$\begin{aligned} \int _{\varGamma } \max _{w\in \mathcal {W}} \mathcal {f}(\bar{\pi }, w) dt < \int _{\varGamma } \Big \lbrace \mathcal {f}(\bar{\eta }, \bar{w})+\bar{\nu }^T G(\bar{\eta },\bar{u})e+\bar{\gamma }^T (H (\bar{\eta },\bar{v})-y_{\alpha })e \Big \rbrace dt. \end{aligned}$$

Since $\max _{w\in \mathcal {W}} \mathcal {f}(\bar{\pi }, w) = \mathcal {f}(\bar{\pi }, \bar{w})$, we have

$$\begin{aligned} \int _{\varGamma } \mathcal {f}(\bar{\pi },\bar{w}) dt < \int _{\varGamma } \Big \lbrace \mathcal {f}(\bar{\eta }, \bar{w})+\bar{\nu }^T G(\bar{\eta },\bar{u})e+\bar{\gamma }^T ( H (\bar{\eta },\bar{v})-y_{\alpha })e \Big \rbrace dt. \end{aligned}$$

The above inequality together with the robust feasibility of $(\bar{x},\bar{c})$ to the problem (MMOPU) yields

$$\begin{aligned} \int _{\varGamma } \Big \lbrace \mathcal {f}(\bar{\pi },\bar{w}) +\bar{\nu }^T G(\bar{\pi },\bar{u})e&+\bar{\gamma }^T ( H (\bar{\pi },\bar{v})-x_{\alpha })e \Big \rbrace dt \\&< \int _{\varGamma } \Big \lbrace \mathcal {f}(\bar{\eta }, \bar{w})+\bar{\nu }^T G(\bar{\eta },\bar{u})e+\bar{\gamma }^T ( H (\bar{\eta },\bar{v})-y_{\alpha })e \Big \rbrace dt. \end{aligned}$$

As $\bar{\mu }^T>0$ and $\bar{\mu }^T e=1$, therefore, the above inequality can be written as

$$\begin{aligned} \int _{\varGamma } \Big \lbrace \bar{\mu }^T\mathcal {f}(\bar{\pi },\bar{w}) +\bar{\nu }^T&G(\bar{\pi },\bar{u}) +\bar{\gamma }^T ( H (\bar{\pi },\bar{v})-x_{\alpha }) \Big \rbrace dt \nonumber \\&< \int _{\varGamma } \Big \lbrace \mu ^T \mathcal {f}(\bar{\eta }, \bar{w})+\bar{\nu }^T G(\bar{\eta },\bar{u})+\bar{\gamma }^T (H (\bar{\eta },\bar{v})-y_{\alpha }) \Big \rbrace dt. \end{aligned}$$

(7.3.5)

Since $\int _{\varGamma } \bar{\mu }^T\mathcal {f}(\cdot ,\bar{w})dt$, $\int _{\varGamma }\bar{\nu }^T G (\cdot ,\bar{u}) dt$ and $\int _{\varGamma } \bar{\gamma }^T (H (\cdot ,\bar{v})-y_{\alpha }) dt$ are convex at $( \bar{y},\bar{z})\in \mathcal {D_{w}}$, we have

$$\begin{aligned}&\int _{\varGamma } \Big \lbrace \bar{\mu }^T\mathcal {f}( \bar{\pi }, \bar{w})-\bar{\mu }^T\mathcal {f}( \bar{\eta }, \bar{w}) \Big \rbrace dt \geqq \int _{\varGamma } (\bar{x}- \bar{y}) \bar{\mu }^T \mathcal {f}_{x}(\bar{\eta },\bar{w})dt+\int _{\varGamma } ( \bar{c}- \bar{z}) \bar{\mu }^T \mathcal {f}_{c}( \bar{\eta },\bar{w})dt, \end{aligned}$$

(7.3.6)

$$\begin{aligned}&\int _{\varGamma } \Big \lbrace \bar{\nu }^T G(\bar{\pi }, \bar{u}) - \bar{\nu }^T G(\bar{\eta }, \bar{u}) \Big \rbrace dt \geqq \int _{\varGamma } (\bar{x}- \bar{y}) \bar{\nu }^T G_{x}(\bar{\eta }, \bar{u}) dt +\int _{\varGamma } ( \bar{c}-\bar{z}) \bar{\nu }^T G_{c}(\bar{\eta }, \bar{u}) dt, \end{aligned}$$

(7.3.7)

and

$$\begin{aligned} \int _{\varGamma }&\Big \lbrace \bar{\gamma }^T (H (\bar{\pi }, \bar{v})-x_{\alpha })- \bar{\gamma }^T (H (\bar{\eta },\bar{v})- y_{\alpha }) \Big \rbrace dt \nonumber \\&~~~~~~~~~~~~~~~~~ \geqq \int _{\varGamma } (\bar{x}-\bar{y})\Big \{\bar{\gamma }^T H_{x} (\bar{\eta }, \bar{v}) +\bar{\gamma }_{t}\Big \} dt +\int _{\varGamma } (\bar{c}- \bar{z}) \bar{\gamma }^T H_{c}(\bar{\eta },\bar{v}) dt. \end{aligned}$$

(7.3.8)

On adding the inequalities (7.3.6), (7.3.7) and (7.3.8) along with the robust feasibility of $( \bar{y}, \bar{z}, \bar{\mu }, \bar{\nu }, \bar{\gamma },\bar{w},\bar{u},\bar{v})$ to the problem (UWRDP), respectively. We have

$$\begin{aligned} \int _{\varGamma } \Big \lbrace \bar{\mu }^T\mathcal {f}( \bar{\pi },\bar{w}) + \bar{\nu }^T G(\bar{\pi },\bar{u})&+ \bar{\gamma }^T (H (\bar{\pi },\bar{v})- x_{\alpha }) \Big \rbrace dt \\&\geqq \int _{\varGamma } \Big \lbrace \bar{\mu }^T\mathcal {f}( \bar{\eta },\bar{w})+ \bar{\nu }^T G(\bar{\eta },\bar{u})+ \bar{\gamma }^T (H (\bar{\eta },\bar{v})- y_{\alpha }) \Big \rbrace dt, \end{aligned}$$

which contradicts the inequality (7.3.5). This completes the proof. $\square$

We present an example to authenticate Theorem 7.3.0.1.

Example 7.3.0.1

Let $p=2, q=r=1, k=2, \mathcal {W}_{1}=[0,1],\mathcal {W}_{2}=[\frac{1}{2},1], \mathcal {U}=[0, 1], \mathcal {V}^1_{1}=\mathcal {V}^1_{2}=[1,2]$ and $\varGamma _{t_{0},t_{1}}$ is fixed by the diagonally opposite points $t_{0}=(t^1_{0},t^2_{0})$ and $t_{1}=(t^1_{1},t^2_{1})$ in $\mathbb {R}^2$.

We consider the following multi-dimensional multi-objective optimization problem:

$$\begin{aligned} {{\text {(MMOPU1)}}}~~ \min _{(x(\cdot ),c(\cdot ))}&\int _{\varGamma _{0,\frac{1}{4}}} \Big (x^2w_{1}+\frac{5}{4}c, \frac{1}{w_{2}}c^2\Big )~d{t}^1 d{t}^2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber \\ ~~~~ \text {subject to}~&~~~ ux(x-3)\le 0, \\&~~~x_{t^1}=v^1_{1}-c, \\&~~~x_{t^2}=v^1_{2}-c, \\&~~~ x(0,0)=0,~ x\Big (\frac{1}{4},\frac{1}{4}\Big )=\frac{1}{2}, \end{aligned}$$

where $t=(t^{1},t^{2}) \in \varGamma _{0,\frac{1}{4}}.$ Let $(\bar{x}, \bar{c})=(t^1, t^2)$ be a robust feasible solution to the problem (MMOPU1).

The robust counterpart of the multi-dimensional multi-objective optimization problem (MMOPU1) is defined as:

$$\begin{aligned}\text {{(RMMOPU1)}}~~ \min _{(x(\cdot ),c(\cdot ))}&\int _{\varGamma _{0,1}} \Big (\max _{w_{1}\in \mathcal {W}_{1}}\Big (x^2w_{1}+\frac{5}{4}c\Big ), \max _{w_{2}\in \mathcal {W}_{2}}\Big (\frac{1}{w_{2}}c^2\Big )\Big )~d{t}^1 d{t}^2 \\ ~~~~ \text {subject to}~&~~~ ux(x-3)\le 0, ~\forall u\in \mathcal {U}, \\&~~~x_{t^1}=v^1_{1}-c, ~\forall v^1_{1}\in \mathcal {V}^1_{1}, \\&~~~x_{t^2}=v^1_{2}-c, ~\forall v^1_{2}\in \mathcal {V}^1_{2}, \\&~~~ x(0,0)=0,~ x\Big (\frac{1}{4},\frac{1}{4}\Big )=\frac{1}{2}, \end{aligned}$$

where $t=(t^{1},t^{2}) \in \varGamma _{0,\frac{1}{4}}$.

The Wolfe type robust dual problem associated with the problem (MMOPU1) is defined as follows:

$$\begin{aligned} \text {{(UWRDP1)}} \max _{(y(\cdot ),z(\cdot ))}\int _{\varGamma _{0,\frac{1}{4}}}&\Big \{\Big (y^2w_{1}+\frac{5}{4}z\Big )+\nu uy(y-3) +\gamma ^1_{1}(y_{t^1}-v^1_{1}+z)\nonumber \\&~~~~~~+\gamma ^1_{2}(y_{t^2}-v^1_{2}+z),\frac{1}{w_{2}}z^2+\nu uy(y-3) \nonumber \\&~~~~~~ +\gamma ^1_{1}(y_{t^1}-v^1_{1}+z)+\gamma ^1_{2}(y_{t^2}-v^1_{2}+z)\Big \}~dt^1 dt^2 \nonumber \\ \text {subject to}~~~~~&2\mu _{1} w_{1} y+ \nu u(2 y-3)+(\gamma ^1_{1})_{t^1}+( \gamma ^1_{1})_{t^2}=0, \end{aligned}$$

(7.3.9)

$$\begin{aligned}&\frac{5}{4} \mu _{1} +2\frac{1}{ w_{2}} z \mu _{2}- \gamma ^1_{1}- \gamma ^1_{2}=0, \end{aligned}$$

(7.3.10)

$$\begin{aligned}&~ y(0,0)=0,~ y\Big (\frac{1}{4},\frac{1}{4}\Big )=\frac{1}{2}, \end{aligned}$$

(7.3.11)

$$\begin{aligned}&~\mu ^{T}> 0,~ e^T \mu =1,~ e=(1, 1)\in \mathbb {R}^2. \end{aligned}$$

(7.3.12)

The robust counterpart to the problem (UWRDP1) is given as:

$$\begin{aligned} \text {{(RUWRDP1)}} \max _{(y(\cdot ),z(\cdot ), w, u, v)}\int _{\varGamma _{0,\frac{1}{4}}}&\Big \{\Big (y^2w_{1}+\frac{5}{4}z\Big )+\nu uy(y-3) +\gamma ^1_{1}(y_{t^1}-v^1_{1}+z) \\&~~~~~~+\gamma ^1_{2}(y_{t^2}-v^1_{2}+z),\frac{1}{w_{2}}z^2+\nu uy(y-3) \\&~~~~~~ +\gamma ^1_{1}(y_{t^1}-v^1_{1}+z)+\gamma ^1_{2}(y_{t^2}-v^1_{2}+z)\Big \}~dt^1 dt^2 \nonumber \\ \text {subject to}~~~~~&2 \mu _{1} w_{1} y+ \nu u(2 y-3)+(\gamma ^1_{1})_{t^1}+(\gamma ^1_{1})_{t^2}=0,\\&\frac{5}{4} \mu _{1} +2\frac{1}{ w_{2}} z \mu _{2}- \gamma ^1_{1}- \gamma ^1_{2}=0, \\&~ y(0,0)=0,~ y\Big (\frac{1}{4},\frac{1}{4}\Big )=\frac{1}{2}, \\&~\mu ^{T}> 0,~ e^T \mu =1,~ e=(1, 1)\in \mathbb {R}^2, \end{aligned}$$

for all $w\in \mathcal {W}, u\in \mathcal {U}, v\in \mathcal {V}$.

Note that $\mathcal {D_{w}}=\{(y,z)\in \mathscr {X}\times \mathscr {C}:~ \text{ satisfying } \text{ conditions }~ \text {(7.3.9)--(7.3.12)}\}$ is the feasible set to the (UWRDP1). Let $\bar{y}=0, \bar{z}=\frac{1}{2},~ \bar{\mu }=(\bar{\mu }_{1},\bar{\mu }_{2})=(1, 0),~ \bar{\nu }=2,~ \bar{\gamma }=( \bar{\gamma }^1_{1},\bar{\gamma }^1_{2})=(1,\frac{1}{4}),~ \bar{w}=(\bar{w}_{1},\bar{w}_{2})=(1,1),~ \bar{u}=1, ~\bar{v}=(\bar{v}^1_{1}, \bar{v}^1_{2})=(1,2)$. Then $(\bar{y}, \bar{z}, \bar{\mu }, \bar{\nu }, \bar{\gamma }, \bar{w},\bar{u}, \bar{v})$ is a robust feasible solution to (UWRDP1). Further, it can be easily verified that all the involved functionals are convex at $(\bar{y}, \bar{z})\in \mathcal {D_{w}}$. Thus, the following inequality

$$\begin{aligned} \int _{\varGamma }&\mathcal {f}(\bar{\pi }, \bar{w}) dt - \int _{\varGamma } \Big \lbrace \mathcal {f}(\bar{\eta }, \bar{w})+\bar{\nu }^T G(\bar{\eta },\bar{u})e+\bar{\gamma }^T ( H (\bar{\eta },\bar{v})-y_{\alpha }) e\Big \rbrace dt \\&=\int _{\varGamma _{0,\frac{1}{4}}} \Big (\bar{x}^2\bar{w}_{1}+\frac{5}{4}\bar{c}, \frac{1}{\bar{w}_{2}}\bar{c}^2\Big ) dt^1 dt^2- \int _{\varGamma _{0,\frac{1}{4}}} \Big \{\Big (\bar{y}^2\bar{w}_{1}+\frac{5}{4}\bar{z}\Big )+\bar{\nu }\bar{u}\bar{y}(\bar{y}-3) \\&~~~~~~~~~~~~~~~~~+\bar{\gamma }^1_{1}(y_{t^1}-\bar{v}^1_{1}+\bar{z}) +\bar{\gamma }^1_{2}(y_{t^2}-\bar{v}^1_{2}+\bar{z}),\frac{1}{\bar{w}_{2}}\bar{z}^2+\bar{\nu }\bar{u}\bar{y}(\bar{y}-3) \\&~~~~~~~~~~~~~~~~~+\bar{\gamma }^1_{1}(\bar{y}_{t^1}-\bar{v}^1_{1}+\bar{z})+\bar{\gamma }^1_{2}(\bar{y}_{t^2}-\bar{v}^1_{2}+\bar{z})\Big \}~dt^1 dt^2 \\&=\int _{\varGamma _{0,\frac{1}{4}}} \Big ( t^1+t^2+\frac{5}{4}, 1 \Big ) dt^1 dt^2-\int _{\varGamma _{0,\frac{1}{4}}} \Big ( \frac{-5}{4}, \frac{-13}{8} \Big ) dt^1 dt^2 \\&= \Big ( \frac{3}{32}, \frac{1}{16} \Big )- \Big ( \frac{-5}{64}, \frac{-13}{128} \Big ) \\&=\Big ( \frac{11}{64}, \frac{21}{128} \Big ) > 0, \end{aligned}$$

which shows that the duality gap is positive. Hence, Theorem 7.3.0.1 (Weak Robust Duality) is verified.

Now, we shall prove the strong robust duality result which states that duality gap is zero.

Theorem 7.3.0.2

(Strong Robust Duality) Let $(\bar{x},\bar{c})$ be a weak robust efficient solution to the problem (MMOPU). Assume that $\max _{w\in \mathcal {W}} \mathcal {f}(\bar{\pi }, w) =\mathcal {f}(\bar{\pi }, \bar{w})$ and the constraint conditions (for the existence of multiplier) holds for (MMOPU). Then, there exist the scalar vector $\bar{\mu }\in \mathbb {R}^s_{+}$, the piecewise smooth Lagrange multipliers $\bar{\nu }=(\bar{\nu }_{l}(t))\in \mathbb {R}^m_{+}, l=\overline{1,m}$ and $\bar{\gamma }=(\bar{\gamma }^{\tau }_{\alpha }(t)) \in \mathbb {R}^{pq}, \tau =\overline{1,q}, \alpha =\overline{1,p}$ and the uncertain parameters $\bar{u}\in \mathcal {U}, \bar{v} \in \mathcal {V}$ such that $(\bar{x},\bar{c},\bar{\mu },\bar{\nu }, \bar{\gamma },\bar{w},\bar{u},\bar{v})$ is a robust feasible solution to the problem (UWRDP). Further, if the Weak Robust Duality (Theorem 7.3.0.1) holds, then $(\bar{x},\bar{c},\bar{\mu },\bar{\nu }, \bar{\gamma },\bar{w},\bar{u},\bar{v})$ is a weak robust efficient solution to the problem (UWRDP).

Proof

Since $(\bar{x},\bar{c})$ be a weak robust efficient solution to the problem (MMOPU), therefore, by Theorem 7.2.0.1, there exist the scalar vector $\bar{\mu }\in \mathbb {R}^s_{+}$, the piecewise smooth Lagrange multiplies $\bar{\nu }\in \mathbb {R}^m_{+}, \bar{\gamma }\in \mathbb {R}^{pq}$ and the uncertain parameters $\bar{u} \in \mathcal {U},\bar{v} \in \mathcal {V}$ such that the conditions (7.2.1)–(7.2.4) are satisfied at $(\bar{x},\bar{c})$. This shows the robust feasibility of $(\bar{x},\bar{c},\bar{\mu },\bar{\nu },\bar{\gamma },\bar{w},\bar{u},\bar{v})$ to the problem (UWRDP) and the corresponding objective values are equal. If $(\bar{x},\bar{c},\bar{\mu },\bar{\nu },\bar{\gamma },\bar{w},\bar{u},\bar{v})$ is not a weak efficient solution to the problem (UWRDP), then there exists another point $(y , z,\bar{\mu },\bar{\nu },\bar{\gamma },\bar{w},\bar{u},\bar{v})$ such that

$$\begin{aligned} \int _{\varGamma } \Big \lbrace \mathcal {f}(\bar{\pi }, \bar{w})+ \nu ^T G(\bar{\pi },\bar{u})&e+\gamma ^T (H (\bar{\pi }, \bar{v})-y_{\alpha })e \Big \rbrace dt \\&< \int _{\varGamma } \Big \lbrace \mathcal {f}(\eta , \bar{w})+ \nu ^T G(\eta ,\bar{u})e +\gamma ^T (H (\eta ,\bar{v})-y_{\alpha })e \Big \rbrace dt. \end{aligned}$$

From the condition (7.2.3), we get

$$\begin{aligned} \int _{\varGamma } \mathcal {f}(\bar{\pi }, \bar{w}) dt < \int _{\varGamma } \Big \lbrace \mathcal {f}(\eta , \bar{w})+ \nu ^T G(\eta ,\bar{u})e +\gamma ^T (H (\eta ,\bar{v})-y_{\alpha })e \Big \rbrace dt. \end{aligned}$$

Since $\max _{w\in \mathcal {W}}\mathcal {f}(\pi , w)=\mathcal {f}(\pi , \bar{w})$, we have

$$\begin{aligned} \int _{\varGamma } \max _{w\in \mathcal {W}}\mathcal {f}(\bar{\pi }, w) dt < \int _{\varGamma } \Big \lbrace \mathcal {f}(\eta , \bar{w})+ \nu ^T G(\eta ,\bar{u})e +\gamma ^T (H (\eta ,\bar{v})-y_{\alpha })e \Big \rbrace dt, \end{aligned}$$

which contradict the Weak Robust Duality (Theorem 7.3.0.1). Hence, $(\bar{x},\bar{c},\bar{\mu },\bar{\nu },\bar{\gamma },\bar{w},\bar{u},\bar{v})$ is a weak efficient solution to the problem (UWRDP). $\square$

Theorem 7.3.0.3

(Strict Converse Robust Duality) Let $(\bar{x},\bar{c}, \bar{\mu }, \bar{\nu }, \bar{\gamma },\bar{w},\bar{u},\bar{v})$ be a robust feasible solution to the problem (UWRDP). Assume that $\max _{w\in \mathcal {W}}\mathcal {f}(\pi , w)=\mathcal {f}(\pi , \bar{w})$ and $\int _{\varGamma }\bar{\mu }^T \mathcal {f}(\cdot ,\bar{w})dt, \int _{\varGamma } \bar{\nu }^T G(\cdot ,\bar{u}) dt$ and $\int _{\varGamma } \{\bar{\gamma }^T H (\cdot , \bar{v})- y_{\alpha } \} dt$ are strict convex at $(\bar{y}, \bar{z})$. If $(\bar{x},\bar{c})\in \mathcal {D}$ such that $\int _{\varGamma }\mathcal {f}(\bar{\pi }, \bar{w})dt=\int _{\varGamma }\mathcal {f}(\bar{\eta }, \bar{w})dt$, then $(\bar{x},\bar{c})$ is a weak robust efficient solution to the problem (MMOPU).

Proof

Since $(\bar{x},\bar{c}, \bar{\mu }, \bar{\nu }, \bar{\gamma },\bar{w},\bar{u},\bar{v})$ is a robust feasible solution to the problem (UWRDP), on multiplying the inequality 8d1) and (7.2.2) by $(\hat{x}- \bar{y})$ and $(\hat{c}- \bar{z})$, respectively, and then integrate them, we get

$$\begin{aligned} \int _{\varGamma } (\hat{x}- \bar{y})\Big \lbrace&\bar{\mu }^T \mathcal {f}_{x}(\bar{\eta }, \bar{w})+ \bar{\nu }^T G_{x}( \bar{\eta },\bar{u}) + \bar{\gamma }^T H_{x}(\bar{\eta }, \bar{v}) + \bar{\gamma }_{t} \Big \rbrace dt\nonumber \\&+\int _{\varGamma } (\hat{c}-\bar{z})\Big \lbrace \bar{\mu }^T \mathcal {f}_{c}(\bar{\eta }, \bar{w})+ \bar{\nu }^T G_{c}( \bar{\eta }, \bar{u})+ \bar{\gamma }^T H_{c}(\bar{\eta }, \bar{v})\Big \rbrace dt = 0. \end{aligned}$$

(7.3.13)

We proceed by contradiction and assume that $(\bar{x},\bar{c})$ is not a weak robust efficient solution to the problem (MMOPU), then there exists $(\hat{x},\hat{c})\in \mathcal {D}$ such that

$$\begin{aligned} \int _{\varGamma }&\max _{w\in \mathcal {W}}\mathcal {f}(\hat{\pi }, w)dt < \int _{\varGamma } \max _{w\in \mathcal {W}} \mathcal {f}(\bar{\pi }, w)dt. \end{aligned}$$

Since $\max _{w\in \mathcal {W}}\mathcal {f}(\pi , w)=\mathcal {f}(\pi , \bar{w})$, therefore

$$\begin{aligned} \int _{\varGamma }&\mathcal {f}(\hat{\pi }, \bar{w})dt < \int _{\varGamma } \mathcal {f}(\bar{\pi }, \bar{w})dt. \end{aligned}$$

By assumption $\int _{\varGamma }\mathcal {f}(\bar{\pi }, \bar{w})dt=\int _{\varGamma }\mathcal {f}(\bar{\eta }, \bar{w})dt$. Therefore, the above inequality yields

$$\begin{aligned} \int _{\varGamma }&\mathcal {f}(\hat{\pi }, \bar{w})dt < \int _{\varGamma } \mathcal {f}(\bar{\eta }, \bar{w})dt. \end{aligned}$$

Since $\bar{\mu }>0\in \mathbb {R}^s_{+}$, therefore

$$\begin{aligned} \int _{\varGamma }&\bar{\mu }^T\mathcal {f}(\hat{\pi }, \bar{w})dt < \int _{\varGamma } \bar{\mu }^T \mathcal {f}(\bar{\eta }, \bar{w})dt. \end{aligned}$$

(7.3.14)

On the other hand, from the assumption $\int _{\varGamma } \bar{\mu }^T \mathcal {f}(\cdot ,\bar{w})dt$ is strict convex at $(\bar{y},\bar{z})$, we have

$$\begin{aligned} \int _{\varGamma } \Big \lbrace \bar{\mu }^T \mathcal {f}(\hat{\pi },\bar{w})- \bar{\mu }^T\mathcal {f}( \bar{\eta },\bar{w}) \Big \rbrace dt > \int _{\varGamma } (\hat{x}- \bar{y}) \bar{\mu }^T \mathcal {f}_{x}(\bar{\eta }, \bar{w})dt+\int _{\varGamma } (\hat{c}-\bar{z}) \bar{\mu }^T \mathcal {f}_{c}(\bar{\eta }, \bar{w})dt, \end{aligned}$$

which together with the inequality (7.3.14), gives

$$\begin{aligned} \int _{\varGamma } (\hat{x}- \bar{y})\bar{\mu }^T \mathcal {f}_{x}(\bar{\eta }, \bar{w})dt+\int _{\varGamma } (\hat{c}-\bar{z})\bar{\mu }^T \mathcal {f}_{c}(\bar{\eta }, \bar{w})dt < 0. \end{aligned}$$

(7.3.15)

Again, by assumption $\int _{\varGamma }\bar{\nu }^T G (\cdot , \bar{u}) dt$ is convex at $(\bar{y},\bar{z})$, we get

$$\begin{aligned} \int _{\varGamma } \Big \lbrace \bar{\nu }^T G(\hat{\pi }, \bar{u}) - \bar{\nu }^T G(\bar{\eta }, \bar{u}) \Big \rbrace dt > \int _{\varGamma } (\hat{x}- \bar{y}) \bar{\nu }^T G_{x}(\bar{\eta }, \bar{u}) dt +\int _{\varGamma } (\hat{c}-\bar{z}) \bar{\nu }^T G_{c}( \bar{\eta }, \bar{u}) dt. \end{aligned}$$

(7.3.16)

Since $(\hat{x}, \hat{c})$ and $(\bar{x},\bar{c},\bar{\nu }, \bar{\gamma },\bar{w},\bar{u},\bar{v})$ are the robust feasible solutions to the problem (MMOPU) and (UWRDP), respectively, we obtain

$$\begin{aligned} \int _{\varGamma }\bar{\nu }^T G(\hat{\pi }, \bar{u}) dt - \int _{\varGamma } \bar{\nu }^T G(\bar{\eta }, \bar{u}) dt \leqq 0, \end{aligned}$$

which along with the inequality (7.3.16), gives

$$\begin{aligned} \int _{\varGamma } (\hat{x}- \bar{y}) \bar{\nu }^T G_{x}( \bar{\eta },\bar{u}) dt +\int _{\varGamma } ( \hat{c}-\bar{z}) \bar{\nu }^T G_{c}( \bar{\eta }, \bar{u}) dt < 0. \end{aligned}$$

(7.3.17)

Similarly, $\int _{\varGamma } \{ \bar{\gamma }^T H (\cdot , \bar{v})-y_{\alpha } \} dt$ is also convex at $(\bar{y},\bar{z})$. The robust feasible solutions $(\hat{x}, \hat{c})$ and $(\bar{x},\bar{c}, \bar{\mu }, \bar{\nu }, \bar{\gamma },\bar{w},\bar{u},\bar{v})$ to the problem (MMOPU) and (UWRDP), respectively, yields

$$\begin{aligned} \int _{\varGamma } (\hat{x}-\bar{y})\Big \lbrace \bar{\gamma }^T H_{x}(\bar{\eta }, \bar{v}) + \bar{\gamma }_{t}\Big \rbrace dt +\int _{\varGamma } (\hat{c}-\bar{z})\bar{\gamma }^T H_{c}(\bar{\eta },\bar{v}) dt < 0. \end{aligned}$$

(7.3.18)

On adding the inequalities (7.3.15), (7.3.17) and (7.3.18), we obtain the following inequality

$$\begin{aligned} \int _{\varGamma }&(\hat{x}- \bar{y})\Big \lbrace \bar{\mu }^T \mathcal {f}_{x}(\bar{\eta }, \bar{w})+ \bar{\nu }^T G_{x}( \bar{\eta },\bar{u}) +\bar{\gamma }^T H_{x}(\bar{\eta }, \bar{v}) + \bar{\gamma }_{t} \Big \rbrace dt\\&~~~~ +\int _{\varGamma } (\hat{c}-\bar{z})\Big \lbrace \bar{\mu }^T \mathcal {f}_{c}(\bar{\eta }, \bar{w})+ \bar{\nu }^T G_{c}( \bar{\eta }, \bar{u})+ \bar{\gamma }^T H_{c}(\bar{\eta }, \bar{v})\Big \rbrace dt < 0, \end{aligned}$$

which contradicts the inequality (7.3.13). This completes the proof. $\square$

7.4 Mond-Weir Type Robust Dual Problem

In this section, we formulate the Mond-Weir type robust dual problem for the considered multi-dimensional multi-objective optimization problem with data uncertainty in the objective and constraint functionals (MMOPU) as follows:

$$\begin{aligned} \text {(UMWRDP)}~~ \max _{(y(\cdot ),z(\cdot ))}&\int _{\varGamma _{t_{0},t_{1}}} \mathcal {f}(t,y(t),z(t),w)dt \nonumber \\ ~~~ \text {subject to}~&~~~~ \mu ^T\mathcal {f}_{x} (t,y(t),z(t),w)+ \nu ^T G_{x} (t,y(t),z(t),u)\nonumber \\&~~~~~~~~~~~~~~~~~~~~~~~~~~+ \gamma ^T H_{x} (t,y(t),z(t),v)+ \gamma _{t}=0, \end{aligned}$$

(7.4.1)

$$\begin{aligned}&~~~~\mu ^T \mathcal {f}_{c}(t,y(t),z(t),w)+ \nu ^T G_{c} (t,y(t),z(t),u)\nonumber \\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+ \gamma ^T H_{c}(t,y(t),z(t),v)=0, \end{aligned}$$

(7.4.2)

$$\begin{aligned}&~~~~ \bar{\nu }^T G(t,y(t),z(t),u)\geqq 0, \end{aligned}$$

(7.4.3)

$$\begin{aligned}&~~~~ y_{\alpha }= H (t,y(t),z(t),v), \end{aligned}$$

(7.4.4)

$$\begin{aligned}&~~~~y(t_{0})=x_{0},~ y(t_{1})=x_{1}, \end{aligned}$$

(7.4.5)

$$\begin{aligned}&~~~~ \mu \in \mathbb {R}^s_{+},~ \mu ^T e=1, \nu \in \mathbb {R}^{m}_{+}, \gamma \in \mathbb {R}^{pq}. \end{aligned}$$

(7.4.6)

The associated robust counterpart to the problem (UMWRDP) is given as follows:

$$\begin{aligned} \text {(RUMWRDP)}~~~~~~ \max _{(y(\cdot ),z(\cdot ),w)}&\int _{\varGamma _{t_{0},t_{1}}} \mathcal {f}(t,y(t),z(t),w)dt \\ ~~~ \text {subject to}~&~~~~ \mu ^T \mathcal {f}_{x} (t,y(t),z(t),w)+ \nu ^T G_{x} (t,y(t),z(t),u)\\&~~~~~~~~~~~~~~~~~~~~~~~~+ \gamma ^T H_{x} (t,y(t),z(t),v)+ \gamma _{t}=0, \\&~~~~\mu ^T \mathcal {f}_{c}(t,y(t),z(t),w)+ \nu ^T G_{c} (t,y(t),z(t),u)\\ {}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~+ \gamma ^T H_{c}(t,y(t),z(t),v)=0, \\&~~~~ \nu ^T G (t,y(t),z(t),u)\geqq 0, \\&~~~~y_{\alpha }= H (t,y(t),z(t),v),\\&~~~~y(t_{0})=x_{0}, y(t_{1})=x_{1}, \\&~~~~\mu ^T e=1, \mu \in \mathbb {R}^s_{+},\nu \in \mathbb {R}^{m}_{+}, \gamma \in \mathbb {R}^{pq}, \end{aligned}$$

for all $w \in \mathcal {W}, u\in \mathcal {U}, v\in \mathcal {V}.$

We denote $\mathcal {D_{mw}}=\lbrace (y,z) \in \mathscr {X}\times \mathscr {C} :~\text{ satisfying } \text{ conditions }~ (7.4.1){-}(7.4.6) \rbrace$ the set of all feasible solutions to (RUWRDP) and we say that it is the robust feasible solution set to the problem (UWRDP).

Now, we establish the robust duality results (weak and strong robust duality results) for (MMOPU) and (UMWRDP) under the convexity hypotheses.

Theorem 7.4.0.1

(Weak Robust Duality) Let $(\bar{x},\bar{c})$ and $(\bar{y}, \bar{z}, \bar{\mu }, \bar{\nu }, \bar{\gamma }, \bar{w}, \bar{u}, \bar{v})$ be the robust feasible solutions to the problem (MMOPU) and (UMWRDP), respectively. Assume that $\max _{w\in \mathcal {W}} \mathcal {f}(\bar{\pi }, w) =\mathcal {f}(\bar{\pi }, \bar{w})$. Further, if $\int _{\varGamma } \bar{\mu }^T \mathcal {f}(\cdot ,\bar{w})dt, \int _{\varGamma } \bar{\nu }^T G(\cdot ,\bar{u}) dt,$ and $\int _{\varGamma } \bar{\gamma }^T \{ H (\cdot , \bar{v})- y_{\alpha } \} dt$ are convex at $(\bar{y},\bar{z})$, then the following inequality cannot hold

$$\begin{aligned} \int _{\varGamma } \max _{w\in \mathcal {W}}\mathcal {f}(\bar{\pi },w) dt < \int _{\varGamma } \mathcal {f}(\bar{\eta },\bar{w}) dt. \end{aligned}$$

Proof

Contrary to the result, we assume that

$$\begin{aligned} \int _{\varGamma } \max _{w\in \mathcal {W}} \mathcal {f}(\bar{\pi },w) dt < \int _{\varGamma } \mathcal {f}(\bar{\eta }, \bar{w})dt. \end{aligned}$$

Since $\max _{w\in \mathcal {W}} \mathcal {f}(\bar{\pi }, w) = \mathcal {f}(\bar{\pi }, \bar{w})$, we have

$$\begin{aligned} \int _{\varGamma } \mathcal {f}(\bar{\pi }, \bar{w}) dt < \int _{\varGamma } \mathcal {f}(\bar{\eta }, \bar{w})dt. \end{aligned}$$

(7.4.7)

By assumptions $\int _{\varGamma } \bar{\mu }^T \mathcal {f}(\cdot ,\bar{w})dt$, $\int _{\varGamma }\bar{\nu }^T G (\cdot , \bar{u}) dt$ and $\int _{\varGamma } \bar{\gamma }^T (H (\cdot , \bar{v})-y_{\alpha }) dt$ are convex at $(\bar{y}, \bar{z})$. Then, we have

$$\begin{aligned}&\int _{\varGamma } \Big \lbrace \bar{\mu }^T \mathcal {f}(\bar{\pi }, \bar{w})-\bar{\mu }^T\mathcal {f}(\bar{\eta }, \bar{w}) \Big \rbrace dt \geqq \int _{\varGamma } ( \bar{x}-\bar{y})\bar{\mu }^T \mathcal {f}_{x}( \bar{\eta }, \bar{w})dt+\int _{\varGamma } (\bar{c}- \bar{z}) \bar{\mu }^T\mathcal {f}_{c}(\bar{\eta }, \bar{w})dt, \end{aligned}$$

(7.4.8)

$$\begin{aligned}&\int _{\varGamma } \Big \lbrace \bar{\nu }^T G(\bar{\pi }, \bar{u}) - \bar{\nu }^T G(\bar{\eta }, \bar{u}) \Big \rbrace dt \geqq \int _{\varGamma } (\bar{x}-\bar{y}) \bar{\nu }^T G_{x}(\bar{\eta }, \bar{u}) dt +\int _{\varGamma } (\bar{c}- \bar{z}) \bar{\nu }^T G_{c}(\bar{\eta }, \bar{u}) dt, \end{aligned}$$

(7.4.9)

$$\begin{aligned}&\text{ and } \nonumber \\&\int _{\varGamma } \Big \lbrace (\bar{\gamma }^T H ( \bar{\pi }, \bar{v})- x_{\alpha })-( \bar{\gamma }^T H (\bar{\eta }, \bar{v})- y_{\alpha }) \Big \rbrace dt \nonumber \\&~~~~~~~~~~~~~~~~~~~ \geqq \int _{\varGamma } (\bar{x}- \bar{y})\Big \lbrace \bar{\gamma }^T H_{x}(\bar{\eta }, \bar{v}) + \bar{\gamma }_{t}\Big \rbrace dt +\int _{\varGamma } ( \bar{c}-\bar{z}) \bar{\gamma }^T H_{c}(\bar{\eta }, \bar{v}) dt. \end{aligned}$$

(7.4.10)

On adding the inequalities (7.4.8), (7.4.9) and (7.4.10) along with the robust feasibility of $(\bar{x},\bar{c})$ and $( \bar{y}, \bar{z}, \bar{\mu }, \bar{\nu }, \bar{\gamma }, \bar{w}, \bar{u}, \bar{v})$ to the problem (MMOPU) and (UMWRDP), respectively, we have

$$\begin{aligned} \int _{\varGamma } \mathcal {f}( \bar{\pi }, \bar{w}) dt \geqq \int _{\varGamma } \mathcal {f}( \bar{\eta }, \bar{w}) dt, \end{aligned}$$

which contradicts the inequality (7.4.7). This completes the proof. $\square$

We present an example to authenticate Theorem 7.4.0.1.

Example 7.4.0.1

Let $p=2, q=r=1, k=2, \mathcal {W}_{1}=\mathcal {W}_{2}=[0,1], \mathcal {U}=[3, 4], \mathcal {V}^1_{1}=\mathcal {V}^1_{2}=[0,2]$ and $\varGamma _{t_{0},t_{1}}$ is fixed by the diagonally opposite points $t_{0}=(t^1_{0},t^2_{0})$ and $t_{1}=(t^1_{1},t^2_{1})$ in $\mathbb {R}^2$.

We consider the following multi-dimensional multi-objective optimization problem:

$$\begin{aligned} {{\text {(MMOPU2)}}}~~ \min _{(x(\cdot ),c(\cdot ))}&\int _{\varGamma _{0,1}} \Big (x^2+c+w_{1}, x+w_{2}c^2+1 \Big )~d{t}^1 d{t}^2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber \\ ~~~~ \text {subject to}~&~~~ u+x(x-4)\le 0, \\&~~~x_{t^1}=2-v^1_{1}c, \\&~~~x_{t^2}=2-v^1_{2}c, \\&~~~ x(0,0)=1,~ x(1,1)=2, \end{aligned}$$

where $t=(t^{1},t^{2}) \in \varGamma _{0,1}.$

The robust counterpart of the multi-dimensional multi-objective optimization problem (MMOPU2) is defined as follows:

$$\begin{aligned}\text {{(RMMOPU2)}}~~ \min _{(x(\cdot ),c(\cdot ))}&\int _{\varGamma _{0,1}} \Big (\max _{w_{1}\in \mathcal {W}_{1}}(x^2+c+w_{1}), \max _{w_{2}\in \mathcal {W}_{2}}(x+w_{2}c^2+1)\Big )~d{t}^1 d{t}^2 \\ ~~~~ \text {subject to}~&~~~ ux(x-3)\le 0, ~\forall u\in \mathcal {U}, \\&~~~x_{t^1}=2-v^1_{1}c, ~\forall v^1_{1}\in \mathcal {V}^1_{1}, \\&~~~x_{t^2}=2-v^1_{2}c, ~\forall v^1_{2}\in \mathcal {V}^1_{2}, \\&~~~ x(0,0)=1,~ x(1,1)=2, \end{aligned}$$

where $t=(t^{1},t^{2}) \in \varGamma _{0,1}$. Let $(\bar{x}, \bar{c})=(\frac{1}{2}(t^1+t^2)+1, \frac{3}{2})$ be a robust feasible solution to the problem (MMOPU2).

The Mond-Weir type robust dual problem associated with the problem (MMOPU2) is defined as follows:

$$\begin{aligned} {{\text {(UMWRDP2)}}} \max _{(y(\cdot ),z(\cdot ))}&\int _{\varGamma _{0,1}} \Big (y^2+z+w_{1}, y+w_{2}z^2+1\Big )~dt^1 dt^2 \nonumber \\ \text {subject to}~~~~~&2\mu _{1}y+\mu _{2}+\nu (2y-4)+(\gamma ^1_{1})_{t^1}+(\gamma ^1_{1})_{t^2}=0, \end{aligned}$$

(7.4.11)

$$\begin{aligned}&\mu _{1} +2 w_{2} z\mu _{2}-v^1_{1}\gamma ^1_{1}-v^1_{2} \gamma ^1_{2}=0, \end{aligned}$$

(7.4.12)

$$\begin{aligned}&\nu (u+y(y-4))\geqq 0, \end{aligned}$$

(7.4.13)

$$\begin{aligned}&y_{t^1}=2-v^1_{1}c, \end{aligned}$$

(7.4.14)

$$\begin{aligned}&y_{t^2}=2-v^1_{2}c, \end{aligned}$$

(7.4.15)

$$\begin{aligned}&y(0,0)=1,~ y(1,1)=2. \end{aligned}$$

(7.4.16)

The robust counterpart to the problem (UMWRDP2) is given as follows:

$$\begin{aligned} \text {{(RUMWRDP2)}} \max _{(y(\cdot ),z(\cdot ),w)}&\int _{\varGamma _{0,1}} \Big (y^2+z+w_{1}, y+w_{2}z^2+1\Big )~dt^1 dt^2 \nonumber \\ \text {subject to}~~~~~&2 \mu _{1}y+\mu _{2}+ \nu (2y-4)+(\gamma ^1_{1})_{t^1}+(\gamma ^1_{1})_{t^2}=0,\\&\mu _{1} +2w_{2}z\mu _{2}-v^1_{1} \gamma ^1_{1}-v^1_{2} \gamma ^1_{2}=0, \\&\nu (u+y(y-4))\geqq 0, \\&y_{t^1}=2-v^1_{1}c, \\&y_{t^2}=2-v^1_{2}c, \\&y(0,0)=1,~ y(1,1)=2, \end{aligned}$$

for all $w\in \mathcal {W}, u\in \mathcal {U}, v\in \mathcal {V}$.

Note that $\mathcal {D_{mw}}=\{(y,z)\in \mathscr {X}\times \mathscr {C}:~ \text{ satisfying } \text{ conditions }~ \text {(7.4.11)--(7.4.16)}\}$ is the feasible set to the (UMWRDP2). Let $\bar{y}=1, \bar{z}=1, \bar{\mu }=(\bar{\mu }_{1}, \bar{\mu }_{2})=(1,0),~ \bar{\nu }=1,~\bar{\gamma }=( \bar{\gamma }^1_{1},\bar{\gamma }^1_{2})=(\frac{1}{4}, \frac{1}{4})~, \bar{w}=( \bar{w}_{1},\bar{w}_{2})=(1,1),~ \bar{u}=4,~\bar{v}=(\bar{v}^1_{1}, \bar{v}^1_{2})=(2,2)$. Then $(\bar{y}, \bar{z}, \bar{\mu }, \bar{\nu }, \bar{\gamma }, \bar{w},\bar{u},\bar{v})$ is a robust feasible solution to (UMWRDP2). Further, it can be easily verified that all the involved functionals are convex at $(\bar{y}, \bar{z})\in \mathcal {D}_{mw}$. Thus, the following inequality

$$\begin{aligned} \int _{\varGamma }&\mathcal {f}(\bar{\pi }, \bar{w}) dt - \int _{\varGamma } \mathcal {f}(\bar{\eta }, \bar{w}) dt \\&=\int _{\varGamma _{0,1}} \Big (\bar{x}^2+\bar{c}+\bar{w}_{1}, \bar{x}+\bar{w}_{2}\bar{c}^2+1\Big ) dt^1 dt^2- \int _{\varGamma _{0,1}} \Big (\bar{y}^2+\bar{z}+\bar{w}_{1}, \bar{y}+\bar{w}_{2}\bar{z}^2+1\Big )~dt^1 dt^2 \\&=\int _{\varGamma _{0,1}} \Big (\frac{1}{2}(t^1+t^2)+1\Big )^2+\frac{3}{2}+1, \frac{1}{2}(t^1+t^2)+\Big (\frac{3}{2}\Big )^2+2 \Big ) dt^1 dt^2-\int _{\varGamma _{0,1}} ( 3,3 ) dt^1 dt^2 \\&= \Big ( \frac{115}{24}, \frac{19}{4} \Big )- \Big ( 3,3 \Big ) \\&=\Big ( \frac{43}{24}, \frac{7}{4} \Big ) > 0, \end{aligned}$$

which shows that the duality gap is positive. Hence, Theorem 7.4.0.1 (Weak Robust Duality) is verified.

Theorem 7.4.0.2

(Strong Robust Duality) Let $(\bar{x},\bar{c})$ be a weak robust efficient solution to the problem (MMOPU). Assume that $\max _{w\in \mathcal {W}} \mathcal {f}(\bar{\pi }, w) =\mathcal {f}(\bar{\pi }, \bar{w})$ and the constraint conditions (for the existence of multiplier) holds for (MMOPU). Then, there exist the scalar vector $\bar{\mu }\in \mathbb {R}^s_{+}$, the piecewise smooth Lagrange multipliers $\bar{\nu }=(\bar{\nu }_{l}(t))\in \mathbb {R}^m_{+}, l=\overline{1,m}$ and $\bar{\gamma }=(\bar{\gamma }^{\tau }_{\alpha }(t)) \in \mathbb {R}^{pq}, \tau =\overline{1,q}, \alpha =\overline{1,p}$ and the uncertain parameters $\bar{u}\in \mathcal {U}, \bar{v} \in \mathcal {V}$ such that $(\bar{x},\bar{c},\bar{\mu },\bar{\nu }, \bar{\gamma },\bar{w},\bar{u},\bar{v})$ is a robust feasible solution to the problem (UMWRDP). Further, if the Weak Robust Duality (Theorem 7.4.0.1) holds, then $(\bar{x},\bar{c},\bar{\mu },\bar{\nu }, \bar{\gamma },\bar{w},\bar{u},\bar{v})$ is a weak robust efficient solution to the problem (UMWRDP).

Proof

Since $(\bar{x},\bar{c})$ be a weak robust efficient solution to the problem (MMOPU), therefore, by Theorem 7.2.0.1, there exist the scalar vector $\bar{\mu }\in \mathbb {R}^s_{+}$, the piecewise smooth Lagrange multiplies $\bar{\nu }\in \mathbb {R}^m_{+}, \bar{\gamma }\in \mathbb {R}^{pq}$ and the uncertain $\bar{u} \in \mathcal {U},\bar{v} \in \mathcal {V}$ such that the conditions (7.2.1)–(7.2.4) are satisfied at $(\bar{x},\bar{c})$. This shows the robust feasibility of $(\bar{x},\bar{c},\bar{\mu },\bar{\nu },\bar{\gamma },\bar{w},\bar{u},\bar{v})$ to the problem (UMWRDP) and the corresponding objective values are equal. If $(\bar{x},\bar{c},\bar{\mu },\bar{\nu },\bar{\gamma },\bar{w},\bar{u},\bar{v})$ is not a weak efficient solution to the problem (UMRDP), then there exists another point $(y , z,\bar{\mu },\bar{\nu },\bar{\gamma },\bar{w},\bar{u},\bar{v})$ such that

$$\begin{aligned} \int _{\varGamma } \max _{w\in \mathcal {W}}\mathcal {f}(\bar{\pi }, w)dt < \int _{\varGamma } \mathcal {f}(\bar{\eta }, \bar{w})dt, \end{aligned}$$

which contradict the Weak Robust Duality (Theorem 7.4.0.1). Hence, $(\bar{x},\bar{c},\bar{\mu },\bar{\nu },\bar{\gamma },\bar{w},\bar{u},\bar{v})$ is a weak efficient solution to the problem (UMWRDP). $\square$

7.5 Mixed Type Robust Dual Problem

In this section, we formulate the mixed type robust dual problem for the considered multi-dimensional multi-objective optimization problem with data uncertainty in the objective and constraint functionals (MMOPU) as follows:

$$\begin{aligned} \text {(UMRDP)}~~ \max _{(y(\cdot ),z(\cdot ))}&\int _{\varGamma _{t_{0},t_{1}}} \Big \lbrace \mathcal {f}(t,y(t),z(t),w)+ \nu ^T G(t,y(t),z(t),u)e\nonumber \\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+\gamma ^T (H (t,y(t),z(t),v)-y_{\alpha })e \Big \rbrace dt \nonumber \\ ~~~ \text {subject to}~&~~~~ \mu ^T\mathcal {f}_{x} (t,y(t),z(t),w)+ \nu ^T G_{x} (t,y(t),z(t),u)\nonumber \\&~~~~~~~~~~~~~~~~~~~~~~~~~~+ \gamma ^T H_{x} (t,y(t),z(t),v)+ \gamma _{t}=0, \end{aligned}$$

(7.5.1)

$$\begin{aligned}&~~~~\mu ^T \mathcal {f}_{c}(t,y(t),z(t),w)+ \nu ^T G_{c} (t,y(t),z(t),u)\nonumber \\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+ \gamma ^T H_{c}(t,y(t),z(t),v)=0,\end{aligned}$$

(7.5.2)

$$\begin{aligned}&~~~~ \bar{\nu }^T G(t,y(t),z(t),u)\ge 0, \end{aligned}$$

(7.5.3)

$$\begin{aligned}&~~~~ y_{\alpha }= H (t,y(t),z(t),v), \end{aligned}$$

(7.5.4)

$$\begin{aligned}&~~~~y(t_{0})=x_{0}, y(t_{1})=x_{1}, \end{aligned}$$

(7.5.5)

$$\begin{aligned}&~~~~ \mu ^T e=1,~ \mu \in \mathbb {R}^s_{+},~\nu \in \mathbb {R}^{m}_{+}, \gamma \in \mathbb {R}^{pq}. \end{aligned}$$

(7.5.6)

The associated robust counterpart to the problem (UMRDP) is given as follows:

$$\begin{aligned} \text {(RUMRDP)}~~~~~~ \max _{(y(\cdot ),z(\cdot ),w,u,v)}&\int _{\varGamma _{t_{0},t_{1}}} \Big \lbrace \mathcal {f}(t,y(t),z(t),w)+ \nu ^T G(t,y(t),z(t),u)e\nonumber \\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+\gamma ^T (H (t,y(t),z(t),v)-y_{\alpha })e \Big \rbrace dt \\ ~~~ \text {subject to}~&~~~~ \mu ^T \mathcal {f}_{x} (t,y(t),z(t),w)+ \nu ^T G_{x} (t,y(t),z(t),u)\\ {}&~~~~~~~~~~~~~~~~~~~~~~~~+ \gamma ^T H_{x} (t,y(t),z(t),v)+ \gamma _{t}=0,\\&~~~~\mu ^T \mathcal {f}_{c}(t,y(t),z(t),w)+ \nu ^T G_{c} (t,y(t),z(t),u)\\ {}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~+ \gamma ^T H_{c}(t,y(t),z(t),v)=0, \\&~~~~ \nu ^T G (t,y(t),z(t),u)\ge 0, \\&~~~~y_{\alpha }= H (t,y(t),z(t),v), \\&~~~~y(t_{0})=x_{0}, y(t_{1})=x_{1}, \\&~~~~\mu ^T e=1, ~\mu \in \mathbb {R}^s_{+},~\nu \in \mathbb {R}^{m}_{+}, \gamma \in \mathbb {R}^{pq}, \end{aligned}$$

for all $w \in \mathcal {W}, u\in \mathcal {U}, v\in \mathcal {V}.$

We denote $\mathcal {D_{m}}=\lbrace (y,z) \in \mathscr {X}\times \mathscr {C} :~\text{ satisfying } \text{ conditions }~ \text {(7.5.1)--(7.5.6)} \rbrace$ the set of all feasible solutions to (RUWRDP) and we say that it is the robust feasible solution set to the problem (UWRDP).

Theorem 7.5.0.1

(Weak Robust Duality) Let $(\bar{x},\bar{c})$ and $( \bar{y},\bar{z},\bar{\mu },\bar{\nu },\bar{\gamma },\bar{w},\bar{u},\bar{v})$ be the robust feasible solutions to the problem (MMOPU) and (UMRDP), respectively. Assume that $\max _{w\in \mathcal {W}} \mathcal {f}(\bar{\pi }, w) = \mathcal {f}(\bar{\pi }, \bar{w})$. Further, if $\int _{\varGamma } \bar{\mu }^T \mathcal {f}(\cdot ,\bar{w})dt, \int _{\varGamma } \bar{\nu }^T G(\cdot ,\bar{u}) dt,$ and $\int _{\varGamma } \bar{\gamma }^T \{ H (\cdot , \bar{v})- y_{\alpha } \} dt$ are convex at $(\bar{y},\bar{z})$, then the following inequality cannot hold

$$\begin{aligned} \int _{\varGamma } \max _{w\in \mathcal {W}} \mathcal {f}(\bar{\pi }, w) dt < \int _{\varGamma } \Big \lbrace \mathcal {f}(\bar{\eta }, \bar{w})+\bar{\nu }^T G(\bar{\eta },\bar{u})e+\bar{\gamma }^T ( H (\bar{\eta },\bar{v})-y_{\alpha }) e\Big \rbrace dt. \end{aligned}$$

Proof

Contrary to the result, we assume that

$$\begin{aligned} \int _{\varGamma } \max _{w\in \mathcal {W}} \mathcal {f}(\bar{\pi }, w) dt < \int _{\varGamma } \Big \lbrace \mathcal {f}(\bar{\eta }, \bar{w})+\bar{\nu }^T G(\bar{\eta },\bar{u})e+\bar{\gamma }^T (H (\bar{\eta },\bar{v})-y_{\alpha })e \Big \rbrace dt. \end{aligned}$$

Since $\max _{w\in \mathcal {W}} \mathcal {f}(\bar{\pi }, w) = \mathcal {f}(\bar{\pi }, \bar{w})$, we have

$$\begin{aligned} \int _{\varGamma } \mathcal {f}(\bar{\pi },\bar{w}) dt < \int _{\varGamma } \Big \lbrace \mathcal {f}(\bar{\eta }, \bar{w})+\bar{\nu }^T G(\bar{\eta },\bar{u})e+\bar{\gamma }^T ( H (\bar{\eta },\bar{v})-y_{\alpha })e \Big \rbrace dt. \end{aligned}$$

The above inequality together with the robust feasibility of $(\bar{x},\bar{c})$ to the problem (MMOPU), yields

$$\begin{aligned} \int _{\varGamma } \Big \lbrace \mathcal {f}(\bar{\pi },\bar{w}) +\bar{\nu }^T G(\bar{\pi },\bar{u})e&+\bar{\gamma }^T ( H (\bar{\pi },\bar{v})-x_{\alpha })e \Big \rbrace dt \\&< \int _{\varGamma } \Big \lbrace \mathcal {f}(\bar{\eta }, \bar{w})+\bar{\nu }^T G(\bar{\eta },\bar{u})e+\bar{\gamma }^T ( H (\bar{\eta },\bar{v})-y_{\alpha })e \Big \rbrace dt. \end{aligned}$$

As $\bar{\mu }^T>0$ and $\bar{\mu }^T e=1$, therefore, the above inequality can be written as

$$\begin{aligned} \int _{\varGamma } \bar{\mu }^T\mathcal {f}(\bar{\pi },\bar{w}) +\bar{\nu }^T&G(\bar{\pi },\bar{u}) +\bar{\gamma }^T ( H (\bar{\pi },\bar{v})-x_{\alpha }) \Big \rbrace dt \nonumber \\&< \int _{\varGamma } \Big \lbrace \bar{\mu }^T \mathcal {f}(\bar{\eta }, \bar{w})+\bar{\nu }^T G(\bar{\eta },\bar{u})+\bar{\gamma }^T (H (\bar{\eta },\bar{v})-y_{\alpha }) \Big \rbrace dt. \end{aligned}$$

(7.5.7)

Since $\int _{\varGamma } \bar{\mu }^T\mathcal {f}(\cdot ,\bar{w})dt$, $\int _{\varGamma }\bar{\nu }^T G (\cdot ,\bar{u}) dt$ and $\int _{\varGamma } \bar{\gamma }^T (H (\cdot ,\bar{v})-y_{\alpha }) dt$ are convex at $(\bar{y},\bar{z})$, we have

$$\begin{aligned}&\int _{\varGamma } \Big \lbrace \bar{\mu }^T\mathcal {f}( \bar{\pi }, \bar{w})-\bar{\mu }^T\mathcal {f}( \bar{\eta }, \bar{w}) \Big \rbrace dt \geqq \int _{\varGamma } (\bar{x}- \bar{y}) \bar{\mu }^T \mathcal {f}_{x}(\bar{\eta },\bar{w})dt+\int _{\varGamma } ( \bar{c}- \bar{z}) \bar{\mu }^T \mathcal {f}_{c}( \bar{\eta },\bar{w})dt, \end{aligned}$$

(7.5.8)

$$\begin{aligned}&\int _{\varGamma } \Big \lbrace \bar{\nu }^T G(\bar{\pi }, \bar{u}) - \bar{\nu }^T G(\bar{\eta }, \bar{u}) \Big \rbrace dt \geqq \int _{\varGamma } (\bar{x}- \bar{y}) \bar{\nu }^T G_{x}(\bar{\eta }, \bar{u}) dt +\int _{\varGamma } ( \bar{c}-\bar{z}) \bar{\nu }^T G_{c}(\bar{\eta }, \bar{u}) dt, \end{aligned}$$

(7.5.9)

and

$$\begin{aligned} \int _{\varGamma }&\Big \lbrace \bar{\gamma }^T (H (\bar{\pi }, \bar{v})-x_{\alpha })- \bar{\gamma }^T (H (\bar{\eta },\bar{v})- y_{\alpha }) \Big \rbrace dt \nonumber \\&~~~~~~~~~~~~~~~~~ \geqq \int _{\varGamma } (\bar{x}-\bar{y})\Big \{\bar{\gamma }^T H_{x} (\bar{\eta }, \bar{v}) +\gamma _{t}\Big \} dt +\int _{\varGamma } (\bar{c}- \bar{z}) \bar{\gamma }^T H_{c}(\bar{\eta },\bar{v}) dt. \end{aligned}$$

(7.5.10)

On adding the inequalities (7.5.8), (7.5.9) and (7.5.10) along with the robust feasibility of $( \bar{x}, \bar{c})$ and $( \bar{y}, \bar{z}, \bar{\mu }, \bar{\nu }, \bar{\gamma },\bar{w},\bar{u},\bar{v})$ to the problem (MMOPU) and (UMRDP), respectively. we have

$$\begin{aligned} \int _{\varGamma } \Big \lbrace \bar{\mu }^T\mathcal {f}( \bar{\pi },\bar{w}) + \bar{\nu }^T&G(\bar{\pi },\bar{u})+ \bar{\gamma }^T (H (\bar{\pi },\bar{v})- x_{\alpha }) \Big \rbrace dt \\&\geqq \int _{\varGamma } \Big \lbrace \bar{\mu }^T\mathcal {f}( \bar{\eta },\bar{w})+ \bar{\nu }^T G(\bar{\eta },\bar{u})+ \bar{\gamma }^T (H (\bar{\eta },\bar{v})- y_{\alpha }) \Big \rbrace dt, \end{aligned}$$

which contradicts the inequality (7.5.7). This completes the proof. $\square$

We are considering the Example 7.4.0.1 to authenticate Theorem 7.5.0.1.

Example 7.5.0.1

The mixed type robust dual problem associated with the considered problem (MMOPU2) is given as follows:

$$\begin{aligned} {{\text {(UMRDP2)}}} \max _{(y(\cdot ),z(\cdot ))}&\int _{\varGamma _{0,1}} \Big \{\Big (y^2+z+w_{1}\Big )+\nu (u+y(y-4)) +\gamma ^1_{1}(y_{t^1}-2+v^1_{1}z) \nonumber \\&~~~~~~+\gamma ^1_{2}(y_{t^2}-2+v^1_{2}z),x+w_{2}c^2+1+\nu (u+y(y-4)) \nonumber \\&~~~~~~ +\gamma ^1_{1}(y_{t^1}-2+v^1_{1}z)+\gamma ^1_{2}(y_{t^2}-2+v^1_{2}z)\Big \}~dt^1 dt^2 \nonumber \\ \text {subject to}~~~~~&2\mu _{1}y+\mu _{2}+\nu (2y-4)+(\gamma ^1_{1})_{t^1}+(\gamma ^1_{1})_{t^2}=0, \end{aligned}$$

(7.5.11)

$$\begin{aligned}&\mu _{1} +2w_{2}z\mu _{2}-v^1_{1} \gamma ^1_{1}-v^1_{2} \gamma ^1_{2}=0, \end{aligned}$$

(7.5.12)

$$\begin{aligned}&\nu (u+y(y-4))\geqq 0, \end{aligned}$$

(7.5.13)

$$\begin{aligned}&y_{t^1}=2-v^1_{1}z, \end{aligned}$$

(7.5.14)

$$\begin{aligned}&y_{t^2}=2-v^1_{2}z, \end{aligned}$$

(7.5.15)

$$\begin{aligned}&~ y(0,0)=1,~ y(1,1)=2. \end{aligned}$$

(7.5.16)

The robust counterpart to the problem (UMRDP2) is given as follows:

$$\begin{aligned} \text {{(RUMRDP2)}} \max _{(y(\cdot ),z(\cdot ),w,u,v)}&\int _{\varGamma _{0,1}} \Big \{\Big (y^2+z+w_{1}\Big )+\nu (u+y(y-4)) +\gamma ^1_{1}(y_{t^1}-2+v^1_{1}z) \\&~~~~~~+\gamma ^1_{2}(y_{t^2}-2+v^1_{2}z),x+w_{2}c^2+1+\nu (u+y(y-4)) \\&~~~~~~ +\gamma ^1_{1}(y_{t^1}-2+v^1_{1}z)+\gamma ^1_{2}(y_{t^2}-2+v^1_{2}z)\Big \}~dt^1 dt^2 \nonumber \\ \text {subject to}~~~~~&2\mu _{1}y+\mu _{2}+\nu (2y-4)+(\gamma ^1_{1})_{t^1}+(\gamma ^1_{1})_{t^2}=0,\\&\mu _{1} +2w_{2}z\mu _{2}-v^1_{1} \gamma ^1_{1}-v^1_{2} \gamma ^1_{2}=0,\\&\nu (u+y(y-4))\geqq 0,\\&y_{t^1}=2-v^1_{1}z,\\&y_{t^2}=2-v^1_{2}z,\\&~ y(0,0)=1,~ y(1,1)=2, \end{aligned}$$

for all $w\in \mathcal {W}, u\in \mathcal {U}, v\in \mathcal {V}$.

Note that $\mathcal {D_{m}}=\{(y,z)\in \mathscr {X}\times \mathscr {C}:~ \text{ satisfying } \text{ conditions }~ \text {(7.5.11)--(7.5.16)}\}$ is the feasible set to the (UMRDP2). Let $(\bar{y}, \bar{z}, \bar{\mu }, \bar{\nu }, \bar{\gamma }, \bar{w}, \bar{u}, \bar{v})$ be a robust feasible solution to (UMRDP2). Thus, the following inequality

$$\begin{aligned} \int _{\varGamma }&\mathcal {f}(\bar{\pi }, \bar{w}) dt - \int _{\varGamma } \Big \lbrace \mathcal {f}(\bar{\eta }, \bar{w})+\bar{\nu }^T G(\bar{\eta },\bar{u})e+\bar{\gamma }^T ( H (\bar{\eta },\bar{v})-y_{\alpha }) e\Big \rbrace dt \\&=\int _{\varGamma _{0,1}} \Big \{\bar{x}^2+\bar{c}+\bar{w}_{1}, ~\bar{x}+\bar{w}_{2}\bar{c}^2+1\Big \} dt^1 dt^2- \int _{\varGamma _{0,1}} \Big \{\bar{y}^2+\bar{z}+\bar{w}_{1}+\bar{\nu }(\bar{u}+\bar{y}(\bar{y}-4)) \\&~~~~~~~~~~+\bar{\gamma }^1_{1}(y_{t^1}-2+\bar{v}^1_{1}\bar{z}) +\bar{\gamma }^1_{2}(y_{t^2}-2+\bar{v}^1_{2}\bar{z}),~ \bar{y}+\bar{w}_{2}\bar{z}^2+1+\bar{\nu }(\bar{u}+\bar{y}(\bar{y}-4)) \\&~~~~~~~~~~+\bar{\gamma }^1_{1}(y_{t^1}-2+\bar{v}^1_{1}\bar{z}) +\bar{\gamma }^1_{2}(y_{t^2}-2+\bar{v}^1_{2}\bar{z})\Big \}~dt^1 dt^2 \\&=\int _{\varGamma _{0,1}} \Big (\Big (\frac{1}{2}(t^1+t^2)+1\Big )^2+\frac{3}{2}+1, \frac{1}{2}(t^1+t^2)+\Big (\frac{3}{2}\Big )^2+2 \Big ) dt^1 dt^2-\int _{\varGamma _{0,1}} ( 4,4 ) dt^1 dt^2 \\&= \Big ( \frac{115}{24}, \frac{19}{4} \Big )- \Big ( 4,4 \Big ) \\&=\Big ( \frac{19}{24}, \frac{3}{4} \Big ) > 0, \end{aligned}$$

which shows that the duality gap is positive. Hence, Theorem 7.5.0.1 (Weak Robust Duality) is verified.

Next, we shall prove the strong robust duality result which states that duality gap is zero.

Theorem 7.5.0.2

(Strong Robust Duality) Let $(\bar{x},\bar{c})$ be a weak robust efficient solution to the problem (MMOPU). Assume that $\max _{w\in \mathcal {W}} \mathcal {f}(\bar{\pi }, w) =\mathcal {f}(\bar{\pi }, \bar{w})$ and the constraint conditions (for the existence of multiplier) holds for (MMOPU). Then, there exist the scalar vector $\bar{\mu }\in \mathbb {R}^s_{+}$, the piecewise smooth Lagrange multipliers $\bar{\nu }=(\bar{\nu }_{l}(t))\in \mathbb {R}^m_{+}, l=\overline{1,m}$ and $\bar{\gamma }=(\bar{\gamma }^{\tau }_{\alpha }(t)) \in \mathbb {R}^{pq}, \tau =\overline{1,q}, \alpha =\overline{1,p}$ and the uncertain parameters $\bar{u}\in \mathcal {U}, \bar{v} \in \mathcal {V}$ such that $(\bar{x},\bar{c},\bar{\mu },\bar{\nu }, \bar{\gamma },\bar{w},\bar{u},\bar{v})$ is a robust feasible solution to the problem (UMRDP). Further, if the Weak Robust Duality (Theorem 7.5.0.1) holds, then $(\bar{x},\bar{c},\bar{\mu },\bar{\nu }, \bar{\gamma },\bar{w},\bar{u},\bar{v})$ is a weak robust efficient solution to the problem (UMRDP).

Proof

Since $(\bar{x},\bar{c})$ be a weak robust efficient solution to the problem (MMOPU). Therefore, by Theorem 7.2.0.1, there exist the scalar vector $\bar{\mu }\in \mathbb {R}^s_{+}$, the piecewise smooth Lagrange multiplies $\bar{\nu }\in \mathbb {R}^m_{+}, \bar{\gamma }\in \mathbb {R}^{pq}$ and the uncertain $\bar{u} \in \mathcal {U},\bar{v} \in \mathcal {V}$ such that the conditions (7.2.1)–(7.2.4) are satisfied at $(\bar{x},\bar{c})$. This shows the robust feasibility of $(\bar{x},\bar{c},\bar{\mu },\bar{\nu },\bar{\gamma },\bar{w},\bar{u},\bar{v})$ to the problem (UMRDP) and the corresponding objective values are equal. If $(\bar{x},\bar{c},\bar{\mu },\bar{\nu },\bar{\gamma },\bar{w},\bar{u},\bar{v})$ is not a weak efficient solution to the problem (UMRDP), then there exists another point $(y , z,\bar{\mu },\bar{\nu },\bar{\gamma },\bar{w},\bar{u},\bar{v})$ such that

$$\begin{aligned} \int _{\varGamma } \max _{w\in \mathcal {W}}\mathcal {f}(\bar{\pi }, w)dt < \int _{\varGamma } \mathcal {f}(\bar{\eta }, \bar{w})dt, \end{aligned}$$

which contradict the Weak Robust Duality (Theorem 7.5.0.1). Hence, $(\bar{x},\bar{c},\bar{\mu },\bar{\nu },\bar{\gamma },\bar{w},\bar{u},\bar{v})$ is a weak efficient solution to the problem (UMRDP). $\square$