1 Introduction

A semi-infinite programming (SIP) is an optimization problem in finitely many variables on a feasible set described by infinitely many constraints. There are many applications of SIP in various fields such as robust optimization, Chebyshev approximation, optimal control, robotics, transportation problems, mathematical physics, fuzzy sets, cooperative games, engineering design (see [1, 2]). For basic theory, survey articles on SIP we refer to [3] and for monograph [4].

The notion of convexificators can be seen as a generalization of subdifferentials. Jeyakumar et al. [5] have shown that the Clarke subdifferentials [6], Michel–Penot subdifferentials [7], Ioffe–Mordukhovich subdifferentials [8] and Treiman subdifferentials [9] of a locally Lipschitz real-valued function are convexificators and these known subdifferentials may contain the convex hull of a convexificator. Convexificators are not necessarily compact or convex, unlike some of the subdifferentials which are compact and convex objects. We refer to the recent results [10,11,12,13] and the references therein for more details related to the convexificators.

Usually, generalized convex functions have been introduced in order to weaken the convexity requirements as much as possible to obtain results related to optimization theory. One of the significant generalization of convex function is invex function [14, 15]. The class of invex functions preserves many properties of the class of convex functions and has shown to be very useful in a variety of applications [16]. It is well known that optimality and duality theory provides the foundation of algorithms for a solution of an optimization problem and hence constitutes an important portion in the study of mathematical programming. Wolfe [17] and Mond–Weir [18] dual models have been studied for semi-infinite programming problems [19, 20], mathematical programs with vanishing constraints [21], bi-level problems [22] and mathematical programming problem with equilibrium constraints [23].

The class of mathematical programs with equilibrium constraints is an extension of the class of bilevel programming problems [24,25,26,27], which is also known as mathematical programs with optimization constraints. Mathematical programming problem with equilibrium constraints (MPPEC) plays a vital role in many fields such as engineering design, capacity enhancement in traffic, economic equilibrium, dynamic pricing in telecommunication networks and multilevel games [28,29,30,31]. Many practical problems in these fields have been modeled using the MPPEC formulation. In practice, it is natural that an MPPEC may arise where infinitely many restrictions are present rather than finite many restrictions in finitely many variables. This gives us a motivation to formulate semi-infinite mathematical programming problem with equilibrium constraints (SIMPPEC).

In this paper, we establish sufficient optimality condition for the SIMPPEC and provide a supportive example corresponding to the sufficient optimality condition. We also derive the weak and strong duality theorems relating to the SIMPPEC and two dual models (Wolfe and Mond–Weir) under the framework of convexificators. The organization of this paper is as follows: In Sect. 2, we give some preliminaries, definitions, and results. In Sect. 3, we establish sufficient optimality condition for the SIMPPEC using convexificators. In Sect. 4, we derive weak and strong duality theorems relating to the SIMPPEC and two dual models using \(\partial ^*\)-invex, \(\partial ^*\)-pseudoinvex and \(\partial ^*\)-quasiinvex functions. In Sect. 5, we conclude the results of this paper.

2 Preliminaries

Throughout the paper, \(\mathbb {R}^n\) denotes the n-dimensional Euclidean space. Let C be a nonempty subset of \(\mathbb {R}^n\). The convex hull of C and the convex cone generated by C are denoted by \(co \ C\) and \(cone \ C\), respectively.

The negative polar cone is defined by \( C^- = \{u \in \mathbb {R}^n \vert \ \forall \ x \in C, \ \langle x,u \rangle \leqslant 0\}.\)

Let C be a nonempty subset of \(\mathbb {R}^n\) and \(x \in cl \ C \) (closure of C), then the contingent cone T(x, C) to C at x is defined by

$$\begin{aligned} T(x, C) = \{u \in \mathbb {R}^n \ \vert \ \exists \ t_n \downarrow 0, \ \exists \ u_n \rightarrow u \ \text {such that} \ x + t_n u_n \in C \}. \end{aligned}$$

We consider SIMPPEC in the following form:

$$\begin{aligned} \text {SIMPPEC}&\quad \text {min}\quad \ F(x) \\&\quad \text {s.t.} \quad g(x,t) \leqslant 0, \ \forall \ t \in T ,\ \ h(x) = 0, \\&\quad \Phi (x) \geqslant 0, \ \Psi (x) \geqslant 0, \ \langle \Phi (x), \Psi (x) \rangle = 0, \end{aligned}$$

where the index set T is an infinite compact subset of \(\mathbb {R}^n, \ F: \mathbb {R}^n \rightarrow \mathbb {R}, \ g : \mathbb {R}^n \times T \rightarrow \mathbb {R},\ h: \mathbb {R}^n \rightarrow \mathbb {R}^p,\ \Phi : \mathbb {R}^n \rightarrow \mathbb {R}^l \ \text {and} \ \Psi : \mathbb {R}^n \rightarrow \mathbb {R}^l\) are given functions.

Along the lines of [19] for a given feasible vector \(\tilde{x} \) for the SIMPPEC, we define the following index sets:

$$\begin{aligned}&T_g := T_g(\tilde{x}) := \{t \in T: g(\tilde{x},t)= 0\},&\\ {}&\delta := \delta (\tilde{x}) := \{i = 1,2,\cdots , l:\Phi _i(\tilde{x})= 0, \Psi _i(\tilde{x})> 0\},&\\ {}&\omega := \omega (\tilde{x}) := \{i = 1,2,\cdots , l:\Phi _i(\tilde{x})= 0, \Psi _i(\tilde{x})= 0\},&\\ {}&\kappa := \kappa (\tilde{x}) := \{i = 1,2,\cdots , l:\Phi _i(\tilde{x})> 0, \Psi _i(\tilde{x})= 0\},&\end{aligned}$$

where the set \(\omega \) is known as degenerate set.

Definition 2.1

Let \(F:\mathbb {R}^n\rightarrow \mathbb {R}\cup \left\{ +\infty \right\} \) be an extended real-valued function, \(x\in \mathbb {R}^n\) and let F(x) be finite. Then, the lower and upper Dini directional derivatives of F at x in the direction v are defined, respectively, by

$$\begin{aligned} F_d^{-}(x, v):=\liminf _{t\rightarrow 0^+}\frac{F(x+tv)-F(x)}{t}, \end{aligned}$$

and

$$\begin{aligned} F_d^{+}(x, v):=\limsup _{t\rightarrow 0^+}\frac{F(x+tv)-F(x)}{t}. \end{aligned}$$

Definition 2.2

(see [5]) A function \(F: \mathbb {R}^n \rightarrow \mathbb {R} \cup \{+ \infty \}\) is said to admit an upper convexificator, \(\partial ^* F(x)\) at \(x \in \mathbb {R}^n\) if \(\partial ^* F(x) \subseteq \mathbb {R}^n\) is a closed set and, for every \(v \in \mathbb {R}^n,\)

$$\begin{aligned} F_d^{-}(x, v)\leqslant \sup _{\xi \in \partial ^*F(x)}\left\langle \xi , v\right\rangle . \end{aligned}$$

Definition 2.3

(see [5]) A function \(F: \mathbb {R}^n \rightarrow \mathbb {R} \cup \{+ \infty \}\) is said to admit a lower convexificator, \(\partial _* F(x)\) at \(x \in \mathbb {R}^n\) if \(\partial _* F(x) \subseteq \mathbb {R}^n\) is a closed set and, for every \(v \in \mathbb {R}^n,\)

$$\begin{aligned} F_d^{+}(x, v)\geqslant \inf _{\xi \in \partial _*F(x)}\left\langle \xi , v\right\rangle . \end{aligned}$$

The function F is said to have a convexificator \(\partial ^*F(x)\subseteq \mathbb {R}^n\) at \(x\in \mathbb {R}^n,\) if \(\partial ^*F(x)\) is both upper and lower convexificator of F at x.

Definition 2.4

(see [32]) A function \(F: \mathbb {R}^n \rightarrow \mathbb {R} \cup \{+\infty \}\) is said to admit an upper semi-regular convexificator, \(\partial ^* F(x)\) at \( x \in \mathbb {R}^n\) if \(\partial ^* F(x) \subseteq \mathbb {R}^n\) is a closed set and, for every \(v \in \mathbb {R}^n\)

$$\begin{aligned} F_d^{+}(x, v)\leqslant \sup _{\xi \in \partial ^*F(x)}\left\langle \xi , v\right\rangle . \end{aligned}$$
(2.1)

If equality holds in (2.1), then \(\partial ^* F(x)\) is called an upper regular convexificator of F at x.

Based on the definitions of invex function [16] and generalized invex functions [33], we will introduce the definitions of invex function and generalized invex functions in terms of convexificators.

Definition 2.5

Let \(\eta :\mathbb {R}^n \times \mathbb {R}^n \rightarrow \mathbb {R}^n\) be a kernel function and let \(F: \mathbb {R}^n \rightarrow \mathbb {R} \cup \{+ \infty \}\) be an extended real-valued function, which admit convexificator at \(\tilde{z} \in \mathbb {R}^n\). Then F is said to be

  1. (i)

    \(\partial ^*\)-invex at \(\tilde{z}\) with respect to \(\eta \) if for every \(x \in \mathbb {R}^n,\)

    $$\begin{aligned} F(x) \geqslant F(\tilde{z}) + \langle \xi , \eta (x, \tilde{z}) \rangle , \forall \ \xi \in \partial ^*F(\tilde{z}). \end{aligned}$$
  2. (ii)

    \(\partial ^*\)-pseudoinvex at \( \tilde{z}\) with respect to \(\eta \) if for every \(x \in \mathbb {R}^n\),

    $$\begin{aligned} \exists \ \xi \in \partial ^*F(\tilde{z}), \ \langle \xi , \eta (x, \tilde{z}) \rangle \geqslant 0 \Rightarrow F(x) \geqslant F(\tilde{z}). \end{aligned}$$
  3. (iii)

    \(\partial ^*\)-quasiinvex at \( \tilde{z}\) with respect to \(\eta \) if for every \(x \in \mathbb {R}^n\),

    $$\begin{aligned} F(x) \leqslant F(\tilde{z}) \Rightarrow \langle \xi , \eta (x, \tilde{z}) \rangle \leqslant 0, \forall \ \xi \in \partial ^*F(\tilde{z}). \end{aligned}$$

Remark 1

Based on Definition 2.5, the definition of \(\partial ^*\)-invex function and generalized \(\partial ^*\)-invex functions can also be given in terms of upper semi-regular convexificators.

Pandey and Mishra [34] presented the following notations for SIMPPEC given for MPPEC by Ardali et al. [35]:

$$\begin{aligned}&g = \bigcup _{i=1}^m co \partial ^*g(\tilde{z},t_i), \ \ h = \bigcup _{i=1}^{p} co \partial ^*h_i(\tilde{z}) \cup co \partial ^*(-h_i)(\tilde{z}),&\\ {}&\Phi _\delta = \bigcup \limits _{i \in \delta } co \partial ^*\Phi _i(\tilde{z}) \cup co \partial ^*(-\Phi _i)(\tilde{z}), \ \ \Phi _{\omega } = \bigcup \limits _{i \in \omega } co \partial ^* \Phi _i(\tilde{z}),&\\ {}&\Psi _\kappa = \bigcup \limits _{i \in \kappa } co \partial ^*\Psi _i(\tilde{z}) \cup co \partial ^*(-\Psi _i)(\tilde{z}), \ \ \Psi _{\omega } = \bigcup \limits _{i \in \omega } co \partial ^* \Psi _i(\tilde{z}),&\\ {}&(\Phi \Psi )_{\omega } = \bigcup \limits _{i \in \omega } co \partial ^*(-\Phi _i)(\tilde{z}) \cup co \partial ^*(-\Psi _i)(\tilde{z}),&\\ {}&\Gamma (\tilde{z}):= g^- \cap h^- \cap \Phi _{\delta }^- \cap \Psi _{\kappa }^- \cap (\Phi \Psi )_{\omega }^-,&\end{aligned}$$

where \(t_1, t_2, \cdots , t_{m} \in T_g(\tilde{z}), \ m \leqslant n+1,\) and \(\tilde{z}\) is a feasible point of SIMPPEC.

The following definitions are taken from Pandey and Mishra [34] for SIMPPEC.

Definition 2.6

Let \(\tilde{z}\) be a feasible point of SIMPPEC, and assume that all functions have convexificators considered above at \(\tilde{z}.\) We say that the generalized standard Abadie constraint qualification (GS Abadie CQ) holds at \(\tilde{z}\) if at least one of the dual sets used in the definition of \(\Gamma (\tilde{z})\) is nonzero and

$$\begin{aligned} \Gamma (\tilde{z}) \subset T(C, \tilde{z}). \end{aligned}$$

Definition 2.7

A feasible point \(\tilde{z}\) of SIMPPEC is called the generalized alternatively stationary (GA-stationary) point if there exist \(\tau = (\tau ^g, \tau ^h, \tau ^{\Phi }, \tau ^{\Psi }) \in \mathbb {R}^{k+p+2l}, \gamma \in (\gamma ^h, \gamma ^{\Phi }, \gamma ^{\Psi }) \in \mathbb {R}^{p+2l}\) and \(t_1, t_2, \cdots , t_m \in T_g(\tilde{z}),\ m \leqslant n+1\), such that the following conditions hold:

$$\begin{aligned}&0 \in co \partial ^* F(\tilde{z}) + \sum _{i = 1}^m \tau _i^g co \partial ^*g(\tilde{z},t_i) + \sum _{r=1}^{p} \left[ \tau _r^h co \partial ^*h_r(\tilde{z}) + \gamma _r^h co \partial ^*(-h_r)(\tilde{z})\right] \nonumber \\&\quad + \sum _{i=1}^{l}\left[ \tau _i^{\Phi } co \partial ^*(-\Phi _i)(\tilde{z}) + \tau _i^{\Psi } co \partial ^*(-\Psi _i)(\tilde{z})\right] \nonumber \\&\quad + \sum _{i=1}^{l}\left[ \gamma _i^{\Phi } co \partial ^*(\Phi _i)(\tilde{z}) + \gamma _i^{\Psi } co \partial ^*(\Psi _i)(\tilde{z})\right] , \nonumber \\&\tau _{i}^g \geqslant 0\quad (i=1,2,\cdots ,m ), \quad \tau _r ^h, \ \gamma _r^h \geqslant 0\quad (r= 1,2, \cdots , p), \end{aligned}$$
(2.2)
$$\begin{aligned}&\tau _i^{\Phi }, \tau _i^{\Psi }, \gamma _i^{\Phi }, \gamma _i^{\Psi } \geqslant 0\quad (i= 1,2, \cdots ,l), \end{aligned}$$
(2.3)
$$\begin{aligned}&\tau _\kappa ^{\Phi } = \tau _\delta ^{\Psi }= \gamma _\kappa ^{\Phi }= \gamma _\delta ^{\Psi }=0, \quad \forall \ i \in \omega , \quad \gamma _i^{\Phi }=0 \ \text {or} \ \gamma _i^{\Psi }=0. \end{aligned}$$
(2.4)

Definition 2.8

A feasible point \(\tilde{z}\) of SIMPPEC is called the generalized strong stationary (GS-stationary) point if there exist \(\tau = (\tau ^g, \tau ^h, \tau ^{\Phi }, \tau ^{\Psi }) \in \mathbb {R}^{k+p+2l}, \gamma \in (\gamma ^h, \gamma ^{\Phi }, \gamma ^{\Psi }) \in \mathbb {R}^{p+2l}\) and \(t_1, t_2, \cdots , t_m \in T_g(\tilde{z}),\ m \leqslant n+1\), satisfying conditions (2.2) and (2.3) together with the following conditions:

$$\begin{aligned} \tau _\kappa ^{\Phi } = \tau _\delta ^{\Psi }= \gamma _\kappa ^{\Phi }= \gamma _\delta ^{\Psi }=0, \quad \forall \ i \in \omega , \quad \gamma _i^{\Phi }=0, \quad \gamma _i^{\Psi }=0. \end{aligned}$$

Note The following index sets will be used in Sects. 3 and 4, respectively:

$$\begin{aligned}&\omega _\gamma ^{\Phi } := \{i \in \omega : \gamma _i ^{\Psi } = 0, \gamma _i^{\Phi }> 0\}, \\&\omega _\gamma ^{\Psi } := \{i \in \omega : \gamma _i^{\Psi }> 0, \gamma _i ^{\Phi }= 0 \}, \\&\delta _\gamma ^+ := \{i \in \delta : \gamma _i^{\Phi }> 0 \}, \\&\kappa _\gamma ^+ : = \{i \in \kappa : \gamma _i ^{\Psi }> 0 \}.&\end{aligned}$$

In the next section, we will obtain sufficient optimality condition under generalized invexity assumptions using the notion of convexificators.

3 Optimality Condition

The following result shows that GS-stationarity is a necessary optimality condition for SIMPPEC.

Theorem 3.1

[34] Let \( \tilde{z}\) be a local optimal solution of SIMPPEC. Suppose that F is locally Lipschitz function at \(\tilde{z}\), which admits a bounded upper semi-regular convexificator \(\partial ^*F(\tilde{z})\). Assume also that GS-ACQ holds at \(\tilde{z}\) and that the cone

$$\begin{aligned} \delta = cone \ co \ g + cone \ co \ h + cone \ co \ \Phi _{\delta } + cone \ co \ \Psi _{\kappa } + cone \ co \ (\Phi \Psi )_{\omega } \end{aligned}$$

is closed, then \(\tilde{z}\) is a GS-stationary point.

Corollary 3.1

[34] Let \(\tilde{z}\) be a local optimal solution of SIMPPEC. Suppose that F is locally Lipschitz near \(\tilde{z}\). Assume also that F and effective constraint functions admit bounded upper semi-regular convexificators at \(\tilde{z}\). If GS-ACQ holds at \(\tilde{z}\), then \(\tilde{z}\) is a GS-stationary point.

The following theorem shows that under generalized \(\partial ^*\)-invexity assumptions, GA-stationarity turns into a global sufficient optimality condition.

Theorem 3.2

Let \(\tilde{z}\) be a feasible GA-stationary point of SIMPPEC and assume that F is \(\partial ^*\)-pseudoinvex at \(\tilde{z}\) with respect to the kernel \(\eta \) and \(g(.,t) \ (t \in T_g), \pm h_r\ (r= 1,2, \cdots , p), \ -\Phi _i\ (i \in \delta \cup \omega ), -\Psi _i\ (i \in \omega \cup \kappa ) \) are \(\partial ^*\)-quasiinvex at \(\tilde{z}\) with respect to the common kernel \(\eta .\) If \(\omega _{\gamma }^{\Phi } \cup \omega _{\gamma }^{\Psi } \cup \delta _{\gamma }^+ \cup \kappa _{\gamma }^+ = \varnothing \ then \ \tilde{z}\) is a global optimal solution of SIMPPEC.

Proof

Let us consider that x be any arbitrary feasible point of SIMPPEC. Then for any \(t_i \in T_g(\tilde{z})\).

$$\begin{aligned} g(x,t_i) \leqslant g(\tilde{z}, t_i), \end{aligned}$$

by \(\partial ^*\)-quasiinvexity of \(g(x,t_i) \) at \(\tilde{z}\), it follows that

$$\begin{aligned} \langle \xi _i^g, \eta (x, \tilde{z}) \rangle \leqslant 0, \ \forall \ \xi _i^g \in \partial ^*g(\tilde{z},t_i), \ \forall \ t_i \in T_g(\tilde{z}). \end{aligned}$$
(3.1)

Similarly, we have

$$\begin{aligned}&\Big \langle \zeta _r, \eta (x,\tilde{z}) \Big \rangle \leqslant 0, \quad \forall \ \zeta _r \in \partial ^* h_r(\tilde{z}), \quad \forall \ r= \{1,2, \cdots , p\}, \end{aligned}$$
(3.2)
$$\begin{aligned}&\Big \langle \nu _r, \eta (x,\tilde{z}) \Big \rangle \leqslant 0, \quad \forall \ \nu _r \in \partial ^* (-h_r)(\tilde{z}), \quad \forall \ r= \{1,2, \cdots , p\}, \end{aligned}$$
(3.3)
$$\begin{aligned}&\Big \langle \xi _i^{\Phi }, \eta (x,\tilde{z}) \Big \rangle \leqslant 0, \quad \forall \ \xi _i^{\Phi } \in \partial ^* (-\Phi _i)(\tilde{z}), \quad \forall \ i \in \delta \cup \omega , \end{aligned}$$
(3.4)
$$\begin{aligned}&\Big \langle \xi _i^{\Psi }, \eta (x,\tilde{z}) \Big \rangle \leqslant 0, \quad \forall \ \xi _i^{\Psi } \in \partial ^* (-\Psi _i)(\tilde{z}), \quad \forall \ i \in \omega \cup \kappa . \end{aligned}$$
(3.5)

If \(\omega _\gamma ^{\Phi } \cup \omega _\gamma ^{\Psi } \cup \delta _\gamma ^+ \cup \kappa _\gamma ^+ = \varnothing ,\) multiplying (3.1)–(3.5) by \(\tau _i^g \geqslant 0 \, (i = 1,2, \cdots , m), \ \tau _r^h> 0 \, (r= 1,2,\cdots , p), \, \gamma _r^h> 0 \, (r=1,2,\cdots ,p), \, \tau _i^{\Phi }> 0 \ (i \in \delta \cup \omega ), \, \tau _i^{\Psi } > 0 \ (i \in \omega \cup \kappa ),\) respectively, and adding, we obtain

$$\begin{aligned}&\Bigg \langle \sum _{i = 1 }^m \tau _i^g {\xi _i^g}, \eta (x,\tilde{z}) \Bigg \rangle \leqslant 0, \quad \Bigg \langle \sum _{r=1}^{p}\left[ \tau _r^h {\zeta }_r + \gamma _r^h{\nu }_r \right] , \eta (x,\tilde{z})\Bigg \rangle \leqslant 0, \\&\Bigg \langle \sum \limits _{\delta \cup \omega } \tau _i^{\Phi } {\xi }_i^{\Phi }, \eta (x,\tilde{z}) \Bigg \rangle \leqslant 0, \quad \Bigg \langle \sum \limits _{\omega \cup \kappa } \tau _i^{\Psi } {\xi }_i^{\Psi }, \eta (x,\tilde{z}) \Bigg \rangle \leqslant 0. \end{aligned}$$

Therefore,

$$\begin{aligned} \Bigg \langle \left( \sum _{i = 1}^m\tau _i^g {\xi }_i^g+ \sum _{r=1}^{p}\left[ \tau _r^h{\zeta }_r+ \gamma _r^h {\nu }_r \right] + \sum \limits _{\delta \cup \omega }\tau _i^{\Phi } {\xi }_i^{\Phi } + \sum \limits _{\omega \cup \kappa }\tau _i^{\Psi } {\xi }_i^{\Psi } \right) , \eta (x, \tilde{z}) \Bigg \rangle \leqslant 0, \end{aligned}$$

for all \( {\xi }_i^g \in co \partial ^*g(\tilde{z},t_i), \ {\zeta _r} \in co \partial ^* h_r(\tilde{z}) , \ {\nu }_r \in co \partial ^*(-h_r)(\tilde{z}), \ {\xi }_i^{\Phi } \in co \partial ^*(-\Phi _i)(\tilde{z}) \) and \( {\xi }_i^{\Psi } \in co \partial ^*(-\Psi _i)(\tilde{z}). \) Thus by GA-stationarity of \(\tilde{z}\), we can choose \(\xi \in co \partial ^*F(\tilde{z}),\) such that

$$\begin{aligned} \langle \xi , \eta (x,\tilde{z}) \rangle \geqslant 0. \end{aligned}$$

By \(\partial ^*\)-pseudoinvexity of F at \(\tilde{z}\) with respect to the common kernel \(\eta \), we have \(F(x) \geqslant F(\tilde{z})\) for all feasible points x. Hence \(\tilde{z}\) is a global optimal solution of SIMPPEC. This completes the proof.

The following example illustrates Theorem 3.2.

Example 3.1

Consider the following SIMPPEC:

SIMPPEC

$$\begin{aligned} \text {min} \ F(x)= & {} {\left\{ \begin{array}{ll} x^2\vert cos \frac{\pi }{x}\vert , &{} \quad x \ne 0 ,\\ 0, &{}\quad x = 0 \end{array}\right. }\\ \hbox {s.t.} \quad g(x,t)= & {} - x^2 - [0,1] \leqslant 0,\ \forall \ t \in [0,1] , \\ \theta (x)= & {} \vert x \vert \geqslant 0, \\ \Psi (x)= & {} x^2 \geqslant 0, \\ \langle \theta (x), \Psi (x) \rangle= & {} \langle \vert x \vert , x^2 \rangle = 0. \end{aligned}$$

Here F is \(\partial ^*\)-pseudoinvex at \(\tilde{z} = 0\) with respect to the kernel, \( \eta (x, \tilde{z})=e^x \sin \tilde{z}.\) Further, \(g, -\theta \ \text {and} -\Psi \) are \(\partial ^*\)-quasiinvex at \(\tilde{z} = 0\) with respect to the common kernel, \(\eta (x, \tilde{z}) = e^x \sin \tilde{z}\). The feasible point for the given SIMPPEC is \(\tilde{z} = 0.\) We have \( co \partial ^*F(0) = [-\pi ,\pi ], \ co \partial ^*g(0, t_1) = \{0\}, \ t_1 = 0,\)\(co \partial ^*(-\theta )(0) = [-1,1] \) and \( co \partial ^*(-\Psi )(0) = \{0\}.\) One can easily verify that there exist \(\tau ^g= 1, \tau ^{\theta }=1\) and \(\tau ^{\Psi } = 1\) such that \(\tilde{z} = 0\) is a GA-stationary point and \(\tilde{z} = 0\) is a global optimal solution for the given primal SIMPPEC. Hence the assumptions of the Theorem 3.2, are satisfied.

4 Duality

In this section, we formulate and study a Wolfe-type dual problem for SIMPPEC using \(\partial ^*\)-invexity. We also formulate Mond–Weir-type dual problem and study SIMPPEC using \(\partial ^*\)-invexity and generalized \(\partial ^*\)-invexity assumptions.

The Wolfe-type dual for SIMPPEC is formulated as follows:

$$\begin{aligned} \text {WD}(\tilde{x}) \quad \quad \max \limits _{z, \tau } \left\{ F(z) {+} \sum \limits _{i = 1}^m \tau _i^g g(z, t_i) {+} \sum _{r=1}^{p} \rho _r^h h_r(z) - \sum _{i=1}^{l}\left[ \tau _i^{\Phi } \Phi _i(z) {+} \tau _i^{\Psi } \Psi _i(z)\right] \right\} \end{aligned}$$

s.t.

$$\begin{aligned}&0 \in co \partial ^* F(z) + \sum _{i = 1}^m \tau _i^g co \partial ^*g(z,t_i) + \sum _{r=1}^{p} \left[ \tau _r^h co \partial ^*h_r(z) + \gamma _r^h co \partial ^*(-h_r)(z)\right] \nonumber \\&\quad \quad +\sum _{i=1}^{l}\left[ \tau _i^{\Phi } co \partial ^*(-\Phi _i)(z) + \tau _i^{\Psi } co \partial ^*(-\Psi _i)(z)\right] , \nonumber \\&\tau _{i}^g \geqslant 0 \ (i = 1,2, \cdots , m), \ \tau _r ^h, \ \gamma _r^h \geqslant 0\ (r=1,2, \cdots , p),\nonumber \\&\tau _i^{\Phi }, \tau _i^{\Psi }, \gamma _i^{\Phi }, \gamma _i^{\Psi } \geqslant 0\ (i= 1,2, \cdots ,l),\nonumber \\&\tau _\kappa ^{\Phi } = \tau _\delta ^{\Psi }= \gamma _\kappa ^{\Phi }= \gamma _\delta ^{\Psi }=0, \ \forall \ i \in \omega , \ \gamma _i^{\Phi }=0, \ \gamma _i^{\Psi }=0, \end{aligned}$$
(4.1)

where \(\rho _r^h= \tau _r^h - \gamma _r^h, \tau = (\tau ^g, \tau ^h, \tau ^{\Phi }, \tau ^{\Psi }) \in \mathbb {R}^{k+p+2l}\), \( \gamma = (\gamma ^h, \gamma ^{\Phi }, \gamma ^{\Psi }) \in \mathbb {R}^{p+2l}\) and \(t_1, t_2, \cdots , t_m \in T_g,\ m \leqslant n + 1.\)

Theorem 4.1

(Weak Duality) Let \(\tilde{x}\) be feasible for SIMPPEC, \((z, \tau )\) be feasible for the dual WD and the index sets \(T_g, \delta , \omega , \kappa \) are defined accordingly. Suppose that \(F, \ g(., t) \ (t \in T), \ \pm h_r \ (r=1,2,\cdots ,p), \ -\Phi _i \ (i \in \delta \cup \omega ), \ -\Psi _i \ (i \in \omega \cup \kappa )\) admit bounded upper semi-regular convexificators and are \(\partial ^*\)-invex functions at z,  with respect to the common kernel \(\eta .\) If \(\omega _\gamma ^{\Phi } \cup \omega _\gamma ^{\Psi } \cup \delta _\gamma ^+ \cup \kappa _\gamma ^+ = \varnothing ,\) then for any x feasible for the SIMPPEC, we have

$$\begin{aligned} F(x) \geqslant F(z) + \sum _{i = 1}^m \tau _i^g g(z,t_i) + \sum _{r=1}^{p} \rho _r^h h_r(z) - \sum _{i=1}^{l}\left[ \tau _i^{\Phi } \Phi _i(z) + \tau _i^{\Psi } \Psi _i(z)\right] . \end{aligned}$$

Proof

Suppose that x be any arbitrary feasible point for the SIMPPEC. It follows that

$$\begin{aligned} g(x,t) \leqslant 0, \ \forall \ t \in T\ \text {and} \ h_r(x) = 0, \ \ r= 1,2, \cdots , p. \end{aligned}$$

Since F is invex at z,  with respect to the kernel \(\eta \), we have

$$\begin{aligned} F(x)- F(z) \geqslant \langle \xi , \eta (x,z) \rangle , \ \forall \ \xi \in \partial ^* F(z). \end{aligned}$$
(4.2)

Similarly,

$$\begin{aligned}&g(x,t_i)- g(z,t_i) \geqslant \Big \langle \xi _i^g, \eta (x,z) \Big \rangle , \ \ \ \ \ \forall \ \xi _i^g \in \partial ^* g(z,t_i), \ \forall \ t_i \in T_g(\tilde{x}), \end{aligned}$$
(4.3)
$$\begin{aligned}&h_r(x)- h_r(z) \geqslant \Big \langle \zeta _r, \eta (x,z) \Big \rangle , \ \ \ \ \forall \ \zeta _r \in \partial ^* h_r(z), \ \forall \ r= \{1,2, \cdots , p\}, \end{aligned}$$
(4.4)
$$\begin{aligned}&-h_r(x)+ h_r(z) \geqslant \Big \langle \nu _r, \eta (x,z) \Big \rangle , \ \ \ \ \forall \ \nu _r \in \partial ^* (-h_r)(z), \ \forall \ r= \{1,2, \cdots , p\}, \qquad \end{aligned}$$
(4.5)
$$\begin{aligned}&-\Phi _i(x)+ \Phi _i(z) \geqslant \Big \langle \xi _i^{\Phi }, \eta (x,z) \Big \rangle ,\ \ \ \forall \ \xi _i^{\Phi } \in \partial ^* (-\Phi _i)(z), \ \forall \ i \in \delta \cup \omega , \end{aligned}$$
(4.6)
$$\begin{aligned}&-\Psi _i(x)+ \Psi _i(z) \geqslant \Big \langle \xi _i^{\Psi }, \eta (x,z) \Big \rangle ,\ \ \ \ \forall \ \xi _i^{\Psi } \in \partial ^* (-\Psi _i)(z), \ \forall \ i \in \omega \cup \kappa . \end{aligned}$$
(4.7)

If \(\omega _\gamma ^{\Phi } \cup \omega _\gamma ^{\Psi } \cup \delta _\gamma ^+ \cup \kappa _\gamma ^+ = \varnothing ,\) then multiplying (4.3)–(4.7) by \(\tau _i^g \geqslant 0 \ (i = 1,2, \cdots , m), \ \tau _r^h> 0 \ (r= 1,2,\cdots , p), \ \gamma _r^h> 0 \ (r=1,2,\cdots ,p), \ \tau _i^{\Phi }> 0 \ (i \in \delta \cup \omega ), \ \tau _i^{\Psi } > 0 \ (i \in \omega \cup \kappa ),\) respectively, and adding (4.2)–(4.7), it follows that

$$\begin{aligned}&F(x)- F(z) + \sum _{i=1}^m \tau _i^g g(x,t_i) - \sum _{i=1}^m \tau _i^g g(z,t_i) + \sum _{r=1}^{p}\tau _r^hh_r(x) \\&\quad \quad -\sum _{r=1}^{p}\tau _r^hh_r(z) -\sum _{r=1}^{p}\gamma _r^hh_r(x) \\&\quad \quad +\sum _{r=1}^{p}\gamma _r^hh_r(z)- \sum _{i=1}^{l}\tau _i^{\Phi }\Phi _i(x)+ \sum _{i=1}^{l}\tau _i^{\Phi }\Phi _i(z)\\&\quad \quad - \sum _{i=1}^{l}\tau _i^{\Psi }\Psi _i(x)+ \sum _{i=1}^{l}\tau _i^{\Psi }\Psi _i(z) \\&\geqslant \Bigg \langle \xi + \sum _{i=1}^m\tau _i^g \xi _i^g+ \sum _{r=1}^{p}\left[ \tau _r^h\zeta _r+ \gamma _r^h \nu _r \right] + \sum _{i=1}^{l}\left[ \tau _i^{\Phi }\xi _i^{\Phi } +\tau _i^{\Psi }\xi _i^{\Psi } \right] , \eta (x,z) \Bigg \rangle . \end{aligned}$$

From (4.1), \( \exists \ \tilde{\xi } \in co \partial ^*F(z), \ \tilde{\xi }_i^g \in co \partial ^*g(z,t_i)\ (t_i \in T_g), \ \tilde{\zeta _r} \in co \partial ^* h_r(z) , \ \tilde{\nu }_r \in co \partial ^*(-h_r)(z), \ \tilde{\xi }_i^{\Phi } \in co \partial ^*(-\Phi _i)(z) \ \text {and} \ \tilde{\xi }_i^{\Psi } \in co \partial ^*(-\Psi _i)(z), \) such that

$$\begin{aligned} \tilde{\xi } + \sum _{i=1}^m\tau _i^g \tilde{\xi }_i^g+ \sum _{r=1}^{p}\left[ \tau _r^h\tilde{\zeta }_r+ \gamma _r^h \tilde{\nu }_r \right] + \sum _{i=1}^{l}\left[ \tau _i^{\Phi }\tilde{\xi }_i^{\Phi } +\tau _i^{\Psi }\tilde{\xi }_i^{\Psi } \right] =0. \end{aligned}$$

Therefore,

$$\begin{aligned}&F(x)- F(z) + \sum _{i=1}^m \tau _i^g g(x,t_i) - \sum _{i=1}^m \tau _i^g g(z,t_i) \\&\quad + \sum _{r=1}^{p}\tau _r^hh_r(x) -\sum _{r=1}^{p}\tau _r^hh_r(z) - \sum _{r=1}^{p}\gamma _r^hh_r(x)\\&\quad +\sum _{r=1}^{p}\gamma _r^hh_r(z)- \sum _{i=1}^{l}\tau _i^{\Phi }\Phi _i(x)+ \sum _{i=1}^{l}\tau _i^{\Phi }\Phi _i(z)\\&\quad - \sum _{i=1}^{l}\tau _i^{\Psi }\Psi _i(x)+ \sum _{i=1}^{l}\tau _i^{\Psi }\Psi _i(z) \geqslant 0. \end{aligned}$$

Using the feasibility of x for SIMPPEC, i.e., \(g(x,t_i) \leqslant 0, \ h_r(x) = 0, \ \Phi _i(x) \geqslant 0, \ \Psi _i(x) \geqslant 0\), it follows that

$$\begin{aligned}&F(x) - F(z) - \sum _{i=1}^m \tau _i^g g(z,t_i)-\sum _{r=1}^{p}\tau _r^hh_r(z)+ \sum _{r=1}^{p}\gamma _r^hh_r(z)\\&\quad + \sum _{i=1}^{l}\tau _i^{\Phi }\Phi _i(z)+ \sum _{i=1}^{l}\tau _i^{\Psi }\Psi _i(z) \geqslant 0. \end{aligned}$$

Hence,

$$\begin{aligned} F(x) \geqslant F(z) + \sum _{i=1}^m \tau _i^g g(z,t_i)+ \sum _{r=1}^{p}\rho _r^hh_r(z)-\sum _{i=1}^{l}\left[ \tau _i^{\Phi } \Phi _i(z) + \tau _i^{\Psi } \Psi _i(z)\right] , \end{aligned}$$

where \(\rho _r^h = \tau _r^h - \gamma _r^h.\) This completes the proof.

Theorem 4.2

(Strong Duality) Let \(\tilde{x}\) be a local optimal solution of SIMPPEC and let F be locally Lipschitz near \(\tilde{x}\). Suppose that \(F, \ g(.,t) \ (t \in T), \ \pm h_r(r= 1,2,\cdots , p), -\Phi _i(i \in \delta \cup \omega ), -\Psi _i(i \in \omega \cup \kappa )\) admit bounded upper semi-regular convexificators and are \(\partial ^*\)-invex functions at \(\tilde{x}\) with respect to the common kernel \(\eta \). If GS-ACQ holds at \( \tilde{x},\) then there exists \(\tilde{\tau } = (\tilde{\tau }^g, \tilde{\tau }^h, \tilde{\tau }^{\Phi }, \tilde{\tau }^{\Psi }) \in \mathbb {R}^{k+p+2l},\) such that \((\tilde{x}, \tilde{\tau })\) is an optimal solution of the dual WD and the respective objective values are equal.

Proof

Since, \(\tilde{x}\) is a local optimal solution of SIMPPEC and GS-ACQ is satisfied at \(\tilde{x},\) now, using Corollary 3.1, \( \exists \ \tilde{\tau } = (\tilde{\tau }^g, \tilde{\tau }^h, \tilde{\tau }^{\Phi }, \tilde{\tau }^{\Psi }) \in \mathbb {R}^{k+p+2l}, \tilde{\gamma } \in (\tilde{\gamma }^h,\tilde{\gamma }^{\Phi },\tilde{\gamma }^{\Psi }) \in \mathbb {R}^{p+2l},\) and indices \(t_1, t_2, \cdots , t_m \in T_g(\tilde{x}),\ m \leqslant n + 1,\) such that GS-stationarity conditions for SIMPPEC are satisfied, that is, \( \exists \ \tilde{\xi } \in co \partial ^* F(\tilde{x}), \ \tilde{\xi }_i^g \in co \partial ^* g(\tilde{x},t_i), \ \tilde{\zeta }_r \in co \partial ^* h_r(\tilde{x}), \ \tilde{\nu }_r \in co \partial ^*(-h_r)(\tilde{x}), \ \tilde{\xi }_i^{\Phi } \in co \partial ^*(-\Phi _i)(\tilde{x}) \ \text {and} \ \tilde{\xi }_i^{\Psi } \in co \partial ^*(-\Psi _i)(\tilde{x}),\) such that

$$\begin{aligned}&\tilde{\xi } + \sum _{i=1}^m\tilde{\tau }_i^g \tilde{\xi }_i^g+ \sum _{r=1}^{p}\left[ \tilde{\tau }_r^h\tilde{\zeta }_r+ \tilde{\gamma }_r^h \tilde{\nu }_r \right] + \sum _{i=1}^{l}\left[ \tilde{\tau }_i^{\Phi }\tilde{\xi }_i^{\Phi } +\tilde{\tau }_i^{\Psi }\tilde{\xi }_i^{\Psi } \right] =0,\\&\tilde{\tau }_{i}^g \geqslant 0\ (i = 1,2, \cdots , m), \ \tilde{\tau }_r ^h, \ \tilde{\gamma }_r^h \geqslant 0 \ \ (r= 1,2, \cdots , p), \\&\tilde{\tau }_i^{\Phi }, \tilde{\tau }_i^{\Psi }, \tilde{\gamma }_i^{\Phi }, \tilde{\gamma }_i^{\Psi } \geqslant 0 \ (i= 1,2, \cdots ,l), \\&\tilde{\tau }_\kappa ^{\Phi } = \tilde{\tau }_\delta ^{\Psi }= \tilde{\gamma }_\kappa ^{\Phi }= \tilde{\gamma }_\delta ^{\Psi }=0, \ \forall \ i \in \omega , \ \tilde{\gamma }_i^{\Phi }=0, \ \tilde{\gamma }_i^{\Psi }=0.&\end{aligned}$$

Therefore \((\tilde{x}, \tilde{\tau })\) is feasible for the dual WD. Now, using Theorem 4.1, we obtain

$$\begin{aligned} F(\tilde{x}) \geqslant F(z) + \sum _{i=1}^m \tau _i^g g(z,t_i) + \sum _{r=1}^{p} \rho _r^h h_r(z) - \sum _{i=1}^{l}\left[ \tau _i^{\Phi } \Phi _i(z) + \tau _i^{\Psi } \Psi _i(z)\right] , \end{aligned}$$
(4.8)

where \(\rho _r^h = \tau _r^h - \gamma _r^h,\) for any feasible solution \((z, \tau )\) for the dual WD. Using the feasibility condition of SIMPPEC and dual WD, that is, for \(t_i \in T_g(\tilde{x}), \ g(\tilde{x}, t_i) = 0,\ h_r(\tilde{x}) = 0, (r= 1,2, \cdots ,p), \ \Phi _i(\tilde{x})=0, \forall i \in \delta \cup \omega , \ \text {and} \ \Psi _i(\tilde{x}) = 0, \forall i \in \omega \cup \kappa ,\) then, we have

$$\begin{aligned} F(\tilde{x}) = F(\tilde{x}) + \sum _{i=1}^m \tilde{\tau }_i^g g(\tilde{x}, t_i) + \sum _{r=1}^{p} \tilde{\rho }_r^h h_r(\tilde{x}) - \sum _{i=1}^{l}\left[ \tilde{\tau }_i^{\Phi } \Phi _i(\tilde{x}) + \tilde{\tau }_i^{\Psi } \Psi _i(\tilde{x})\right] , \end{aligned}$$
(4.9)

where \(\tilde{\rho }_r^h = \tilde{\tau }_r^h - \tilde{\gamma }_r^h.\) Using (4.8) and (4.9), it follows that

$$\begin{aligned}&\quad F(\tilde{x}) + \sum _{i=1}^m \tilde{\tau }_i^g g(\tilde{x}, t_i) + \sum _{r=1}^{p} \tilde{\rho }_r^h h_r(\tilde{x}) - \sum _{i=1}^{l}\left[ \tilde{\tau }_i^{\Phi } \Phi _i(\tilde{x}) + \tilde{\tau }_i^{\Psi } \Psi _i(\tilde{x})\right] \\&\geqslant F(z) + \sum _{i=1}^m \tau _i^g g(z,t_i) + \sum _{r=1}^{p} \rho _r^h h_r(z) - \sum _{i=1}^{l}\left[ \tau _i^{\Phi } \Phi _i(z) + \tau _i^{\Psi } \Psi _i(z)\right] . \end{aligned}$$

Hence, \((\tilde{x}, \tilde{\tau })\) is an optimal solution for the dual WD and the respective objective values are equal. This completes the proof.

Now, we formulate the Mond–Weir-type dual problem (MWDP) for SIMPPEC and establish duality theorems using convexificators.

$$\begin{aligned} \text {MWDP} (\tilde{x}) \ \ \ \ \ \ \ \ \ \ \ \ \ \max \limits _{z, \tau }F(z) \end{aligned}$$

s.t.

$$\begin{aligned}&0 \in co \partial ^* F(z) + \sum _{i=1}^m \tau _i^g co \partial ^*g(z,t_i) + \sum _{r=1}^{p} \left[ \tau _r^h co \partial ^*h_r(z) + \gamma _r^h co \partial ^*(-h_r)(z)\right] \nonumber \\&\quad \quad + \sum _{i=1}^{l}\left[ \tau _i^{\Phi } co \partial ^*(-\Phi _i)(z) + \tau _i^{\Psi } co \partial ^*(-\Psi _i)(z)\right] , \nonumber \\&\quad g(z,t_i) \geqslant 0 \ (t_i \in T_g(\tilde{x})), \ \ h_r(z) = 0 \ (r=1,2, \cdots ,p),\ \ \ \ \ \ \ \ \ \nonumber \\&\quad \Phi _i(z) \leqslant 0 \ (i \in \delta \cup \omega ), \ \ \Psi _i(z) \leqslant 0 \ (i \in \omega \cup \kappa ), \nonumber \\&\quad \tau _{i}^g \geqslant 0\ (i= 1,2, \cdots , m),\ \tau _r ^h, \gamma _r^h \geqslant 0 \ (r= 1,2, \cdots , p), \nonumber \\&\quad \tau _i^{\Phi }, \tau _i^{\Psi }, \gamma _i^{\Phi }, \gamma _i^{\Psi } \geqslant 0 \ (i= 1,2, \cdots ,l), \nonumber \\&\quad \tau _\kappa ^{\Phi } = \tau _\delta ^{\Psi }= \gamma _\kappa ^{\Phi }= \gamma _\delta ^{\Psi }=0, \ \forall \ i \in \omega , \gamma _i^{\Phi }=0, \gamma _i^{\Psi }=0, \end{aligned}$$
(4.10)

where \(\tau = (\tau ^g, \tau ^h, \tau ^{\Phi }, \tau ^{\Psi }) \in \mathbb {R}^{k+p+2l}, \ \gamma = (\gamma ^h, \gamma ^{\Phi }, \gamma ^{\Psi }) \in \mathbb {R}^{p+2l} \ \text {and} \ \ t_1, t_2, \cdots , t_m \in T_g(\tilde{x}),\ m \leqslant n + 1.\)

Theorem 4.3

(Weak Duality) Let \(\tilde{x}\) be feasible for SIMPPEC, \((z, \tau )\) be feasible for the MWDP and the index sets \(T_g, \delta , \omega , \kappa \) be defined accordingly. Suppose that \(F, g(.,t) \ (t \in T), \pm h_r(r=1,2,\cdots ,p), -\Phi _i(i \in \delta \cup \omega ), -\Psi _i (i \in \omega \cup \kappa )\) admit bounded upper semi-regular convexificators and are \(\partial ^*\)-invex functions at z,  with respect to the common kernel \(\eta .\) If \(\omega _\gamma ^{\Phi } \cup \omega _\gamma ^{\Psi } \cup \delta _\gamma ^+ \cup \kappa _\gamma ^+ = \varnothing ,\) then for any x feasible for SIMPPEC, we have

$$\begin{aligned} F(x) \geqslant F(z). \end{aligned}$$

Proof

Since F is invex at z,  with respect to the kernel \(\eta ,\) then, we have

$$\begin{aligned} F(x)- F(z) \geqslant \big \langle \xi , \eta (x,z) \big \rangle , \quad \forall \ \xi \in \partial ^* F(z). \end{aligned}$$
(4.11)

Similarly, we have

$$\begin{aligned}&g(x,t_i)- g(z,t_i) \geqslant \Big \langle \xi _i^g, \eta (x,z) \Big \rangle , \quad \forall \ \xi _i^g \in \partial ^* g(z,t_i), \ \forall \ t_i \in T_g, \end{aligned}$$
(4.12)
$$\begin{aligned}&h_r(x)- h_r(z) \geqslant \Big \langle \zeta _r, \eta (x,z) \Big \rangle , \quad \forall \ \zeta _r \in \partial ^* h_r(z), \ \forall \ r= \{1,2, \cdots , p\}, \end{aligned}$$
(4.13)
$$\begin{aligned}&-h_r(x)+ h_r(z) \geqslant \Big \langle \nu _r, \eta (x,z) \Big \rangle , \quad \forall \ \nu _r \in \partial ^* (-h_r)(z), \ \forall \ r= \{1,2, \cdots , p\}, \qquad \qquad \end{aligned}$$
(4.14)
$$\begin{aligned}&-\Phi _i(x)+ \Phi _i(z) \geqslant \Big \langle \xi _i^{\Phi }, \eta (x,z) \Big \rangle , \quad \forall \ \xi _i^{\Phi } \in \partial ^* (-\Phi _i)(z), \ \forall \ i \in \delta \cup \omega , \end{aligned}$$
(4.15)
$$\begin{aligned}&-\Psi _i(x)+ \Psi _i(z) \geqslant \Big \langle \xi _i^{\Psi }, \eta (x,z) \Big \rangle ,\quad \forall \ \xi _i^{\Psi } \in \partial ^* (-\Psi _i)(z), \ \forall \ i \in \omega \cup \kappa . \end{aligned}$$
(4.16)

If \(\omega _\gamma ^{\Phi } \cup \omega _\gamma ^{\Psi } \cup \delta _\gamma ^+ \cup \kappa _\gamma ^+ = \varnothing ,\) multiplying (4.12)–(4.16) by \(\tau _i^g \geqslant 0 \ (i = 1,2, \cdots , m), \tau _r^h> 0 \ (r= 1,2,\cdots , p), \ \gamma _r^h> 0 \ (r=1,2,\cdots ,p), \ \tau _i^{\Phi }> 0 \ (i \in \delta \cup \omega ), \ \tau _i^{\Psi } > 0 \ (i \in \omega \cup \kappa ),\) respectively, and adding (4.11)–(4.16), we obtain

$$\begin{aligned}&F(x)- F(z) + \sum _{i=1}^m \tau _i^g g(x,t_i) - \sum _{i=1}^m \tau _i^g g(z,t_i) \\&\quad \quad + \sum _{r=1}^{p}\tau _r^hh_r(x) -\sum _{r=1}^{p}\tau _r^hh_r(z) - \sum _{r=1}^{p}\gamma _r^hh_r(x)\\&\quad \quad +\sum _{r=1}^{p}\gamma _r^hh_r(z)- \sum _{i=1}^{l}\tau _i^{\Phi }\Phi _i(x)+ \sum _{i=1}^{l}\tau _i^{\Phi }\Phi _i(z)\\&\quad \quad - \sum _{i=1}^{l}\tau _i^{\Psi }\Psi _i(x)+ \sum _{i=1}^{l}\tau _i^{\Psi }\Psi _i(z) \\&\quad \geqslant \Bigg \langle \xi + \sum _{i=1}^m\tau _i^g \xi _i^g+ \sum _{r=1}^{p}\left[ \tau _r^h\zeta _r+ \gamma _r^h \nu _r \right] \\&\quad \quad + \sum _{i=1}^{l}\left[ \tau _i^{\Phi }\xi _i^{\Phi } +\tau _i^{\Psi }\xi _i^{\Psi } \right] , \eta (x,z) \Bigg \rangle . \end{aligned}$$

From (4.10), \( \exists \ \tilde{\xi } \in co \partial ^*F(z), \ \tilde{\xi }_i^g \in co \partial ^*g(z,t_i), \tilde{\zeta }_r \in co \partial ^* h_r(z), \ \tilde{\nu }_r \in co \partial ^*(-h_r)(z), \ \tilde{\xi }_i^{\Phi } \in co \partial ^*(-\Phi _i)(z) \ \text {and} \ \tilde{\xi }_i^{\Psi } \in co \partial ^*(-\Psi _i)(z), \) such that

$$\begin{aligned} \tilde{\xi } + \sum _{i=1}^m\tau _i^g \tilde{\xi }_i^g+ \sum _{r=1}^{p}\left[ \tau _r^h\tilde{\zeta }_r+ \gamma _r^h \tilde{\nu }_r \right] + \sum _{i=1}^{l}\left[ \tau _i^{\Phi }\tilde{\xi }_i^{\Phi } +\tau _i^{\Psi }\tilde{\xi }_i^{\Psi } \right] =0. \end{aligned}$$

Therefore,

$$\begin{aligned}&F(x)- F(z) + \sum _{i=1}^m \tau _i^g g(x,t_i) - \sum _{i=1}^m \tau _i^g g(z,t_i) + \sum _{r=1}^{p}\tau _r^hh_r(x)\\&\quad -\sum _{r=1}^{p}\tau _r^hh_r(z) - \sum _{r=1}^{p}\gamma _r^hh_r(x) \\&\quad +\sum _{r=1}^{p}\gamma _r^hh_r(z)- \sum _{i=1}^{l}\tau _i^{\Phi }\Phi _i(x)+ \sum _{i=1}^{l}\tau _i^{\Phi }\Phi _i(z)\\&\quad - \sum _{i=1}^{l}\tau _i^{\Psi }\Psi _i(x)+ \sum _{i=1}^{l}\tau _i^{\Psi }\Psi _i(z) \geqslant 0. \end{aligned}$$

Using the feasibility of x and z for SIMPPEC and MWDP, respectively, we obtain

$$\begin{aligned} F(x) \geqslant F(z). \end{aligned}$$

This completes the proof.

Theorem 4.4

(Strong Duality) Let \(\tilde{x}\) be a local optimal solution of SIMPPEC and let F be locally Lipschitz near \(\tilde{x}\). Suppose that \(F, \ g(.,t) \ (t \in T), \ \pm h_r \ (r= 1,2,\cdots , p), \ -\Phi _i \ (i \in \delta \cup \omega ), \ -\Psi _i \ (i \in \omega \cup \kappa )\) admit bounded upper semi-regular convexificators and are \(\partial ^*\)-invex functions at \(\tilde{x}\) with respect to the common kernel \(\eta .\) If GS-ACQ holds at \( \tilde{x},\) then there exists \(\tilde{\tau }\), such that \((\tilde{x}, \tilde{\tau })\) is an optimal solution of the MWDP and the respective objective values are equal.

Proof

Since, \(\tilde{x}\) is a local optimal solution of SIMPPEC and the GS-ACQ is satisfied at \(\tilde{x},\) now using Corollary 3.1, \( \exists \ \tilde{\tau } = (\tilde{\tau }^g, \tilde{\tau }^h, \tilde{\tau }^{\Phi }, \tilde{\tau }^{\Psi }) \in \mathbb {R}^{k+p+2l}, \tilde{\gamma } \in (\tilde{\gamma }^h,\tilde{\gamma }^{\Phi },\tilde{\gamma }^{\Psi }) \in \mathbb {R}^{p+2l},\) and indices \(t_1, t_2, \cdots , t_m \in T_g(\tilde{x}), \ m \leqslant n + 1,\) such that the GS-stationarity conditions for SIMPPEC are satisfied, that is, \( \exists \ \tilde{\xi } \in co \partial ^* F(\tilde{x}), \ \tilde{\xi }_i^g \in co \partial ^* g(\tilde{x},t_i), \ \tilde{\zeta }_r \in co \partial ^* h_r(\tilde{x}), \ \tilde{\nu }_r \in co \partial ^*(-h_r)(\tilde{x}), \ \tilde{\xi }_i^{\Phi } \in co \partial ^*(-\Phi _i)(\tilde{x}) \ \text {and} \ \tilde{\xi }_i^{\Psi } \in co \partial ^*(-\Psi _i)(\tilde{x}),\) such that

$$\begin{aligned}&\tilde{\xi } + \sum _{i=1}^m\tilde{\tau }_i^g \tilde{\xi }_i^g+ \sum _{r=1}^{p}\left[ \tilde{\tau }_r^h\tilde{\zeta }_r+ \tilde{\gamma }_r^h \tilde{\nu }_r \right] + \sum _{i=1}^{l}\left[ \tilde{\tau }_i^{\Phi }\tilde{\xi }_i^{\Phi } +\tilde{\tau }_i^{\Psi }\tilde{\xi }_i^{\Psi } \right] =0,\\&\tilde{\tau }_{i}^g \geqslant 0, (i = 1,2, \cdots , m), \ \tilde{\tau }_r ^h, \ \tilde{\gamma }_r^h \geqslant 0 \ \ (r= 1,2, \cdots , p), \\&\tilde{\tau }_i^{\Phi }, \tilde{\tau }_i^{\Psi }, \tilde{\gamma }_i^{\Phi }, \tilde{\gamma }_i^{\Psi } \geqslant 0 \ (i= 1,2, \cdots ,l), \\&\tilde{\tau }_\kappa ^{\Phi } = \tilde{\tau }_\delta ^{\Psi }= \tilde{\gamma }_\kappa ^{\Phi }= \tilde{\gamma }_\delta ^{\Psi }=0, \forall \ i \in \omega , \ \tilde{\gamma }_i^{\Phi }=0, \ \tilde{\gamma }_i^{\Psi }=0.&\end{aligned}$$

Since \(\tilde{x}\) is an optimal solution for SIMPPEC, we have

$$\begin{aligned} \sum _{i=1}^{m}\tilde{\tau }_i^gg(\tilde{x}, t_i) = 0, \quad \sum _{i=1}^{p}\tilde{\tau }_i^hh_i(\tilde{x}) = 0, \quad \sum _{i=1}^{l}\tilde{\tau }_i^{\Phi }\Phi _i(\tilde{x}) = 0, \quad \sum _{i=1}^{l}\tilde{\tau }_i^{\Psi }\Psi _i(\tilde{x}) = 0. \end{aligned}$$

Therefore \((\tilde{x}, \tilde{\tau })\) is feasible for MWDP. By Theorem 4.3, for any feasible \((z, \tau ),\) we have

$$\begin{aligned} F(\tilde{x}) \geqslant F(z). \end{aligned}$$

It follows that \((\tilde{x}, \tilde{\tau })\) is an optimal solution for MWDP and the respective objective values are equal. This completes the proof.

Now, we establish weak and strong duality theorems for SIMPPEC and its MWDP under generalized \(\partial ^*\)-invexity assumptions.

Theorem 4.5

(Weak Duality) Let \(\tilde{x}\) be feasible for SIMPPEC, \((z, \tau )\) be feasible for the MWDP and the index sets \(T_g, \delta , \omega , \kappa \) are defined accordingly. Suppose that F is \(\partial ^*\)-pseudoinvex at z,  with respect to the kernel \(\eta \) and \(g(.,t) \ (t \in T), \ \pm h_r \ (r=1,2,\cdots ,p), \ -\Phi _i \ (i \in \delta \cup \omega ), \ -\Psi _i \ (i \in \omega \cup \kappa )\) admit bounded upper semi-regular convexificators and are \(\partial ^*\)-quasiinvex functions at z,  with respect to the common kernel \(\eta .\) If \(\omega _\gamma ^{\Phi } \cup \omega _\gamma ^{\Psi } \cup \delta _\gamma ^+ \cup \kappa _\gamma ^+ = \varnothing ,\) then for any x feasible for the problem SIMPPEC, we have

$$\begin{aligned} F(x) \geqslant F(z). \end{aligned}$$

Proof

Suppose that for some feasible point x, such that \(F(x) < F(z),\) then by \(\partial ^*\)-pseudoinvexity of F at z, with respect to the kernel \(\eta ,\) we have

$$\begin{aligned} \langle \xi , \eta (x,z) \rangle < 0, \quad \forall \ \xi \in \partial ^*F(z). \end{aligned}$$
(4.17)

From (4.10), \( \exists \, \tilde{\xi } \in co \partial ^*F(z), \, \tilde{\xi }_i^g \in co \partial ^*g(z,t_i), \tilde{\zeta }_r \in co \partial ^* h_r(z), \, \tilde{\nu }_r \in co \partial ^*(-h_r)(z),\) \(\tilde{\xi }_i^{\Phi } \in co \partial ^*(-\Phi _i)(z) \, \text {and} \, \tilde{\xi }_i^{\Psi } \in co \partial ^*(-\Psi _i)(z), \) such that

$$\begin{aligned} - \sum _{i=1}^m\tau _i^g \tilde{\xi }_i^g- \sum _{r=1}^{p}\left[ \tau _r^h\tilde{\zeta }_r+ \gamma _r^h \tilde{\nu }_r \right] - \sum \limits _{\delta \cup \omega }\tau _i^{\Phi } \tilde{\xi }_i^{\Phi } - \sum \limits _{\omega \cup \kappa }\tau _i^{\Psi } \tilde{\xi }_i^{\Psi } \in \partial ^* F(z). \end{aligned}$$
(4.18)

By (4.17), we have

$$\begin{aligned}&\quad \left\langle \left( \sum _{i=1}^m\tau _i^g \tilde{\xi }_i^g+ \sum _{r=1}^{p}\left[ \tau _r^h\tilde{\zeta }_r+ \gamma _r^h \tilde{\nu }_r \right] + \sum \limits _{\delta \cup \omega }\tau _i^{\Phi } \tilde{\xi }_i^{\Phi } + \sum \limits _{\omega \cup \kappa }\tau _i^{\Psi } \tilde{\xi }_i^{\Psi } \right) , \eta (x,z) \right\rangle \nonumber \\&> 0. \end{aligned}$$
(4.19)

For each \(t_i \in T_g, \ g(x,t_i) \leqslant 0 \leqslant g(z,t_i)\). Hence, by \(\partial ^*\)-quasiinvexity, it follows that

$$\begin{aligned} \Big \langle \xi _i^g, \eta (x,z) \Big \rangle \leqslant 0, \quad \forall \ \xi _i^g \in \partial ^* g(z,t_i), \ \forall \ t_i \in T_g. \end{aligned}$$
(4.20)

Similarly, we have

$$\begin{aligned} \Big \langle \zeta _r, \eta (x,z) \Big \rangle \leqslant 0, \quad \forall \ \zeta _r \in \partial ^* h_r(z), \ \forall \ r= \{1,2, \cdots , p\}. \end{aligned}$$
(4.21)

For any feasible point z of the MWDP, and for every \(r, -h_r(z) = h_r(x) = 0\). On the other hand, \(-\Phi _i(x) \leqslant -\Phi _i(z), \forall i \in \delta \cup \omega \), and \(-\Psi _i(x) \leqslant -\Psi _i(z), \forall i \in \omega \cup \kappa \). By \(\partial ^*\)-quasiinvexity, we have

$$\begin{aligned}&\Big \langle \nu _r, \eta (x,z) \Big \rangle \leqslant 0, \quad \forall \ \nu _r \in \partial ^* (-h_r)(z), \ \forall \ r= \{1,2, \cdots , p\}, \end{aligned}$$
(4.22)
$$\begin{aligned}&\Big \langle \xi _i^{\Phi }, \eta (x,z) \Big \rangle \leqslant 0,\quad \forall \ \xi _i^{\Phi } \in \partial ^* (-\Phi _i)(z), \forall \ i \in \delta \cup \omega , \end{aligned}$$
(4.23)
$$\begin{aligned}&\Big \langle \xi _i^{\Psi }, \eta (x,z) \Big \rangle \leqslant 0, \quad \forall \ \xi _i^{\Psi } \in \partial ^* (-\Psi _i)(z), \forall \ i \in \omega \cup \kappa . \end{aligned}$$
(4.24)

From Eqs. (4.20)–(4.24), we have

$$\begin{aligned}&\Big \langle \tilde{\xi }_i^g, \eta (x,z) \Big \rangle \leqslant 0, \quad (i = 1,2, \cdots , m), \\&\Big \langle \tilde{\zeta }_r, \eta (x,z) \Big \rangle \leqslant 0, \quad \Big \langle \tilde{\nu }_r, \eta (x,z) \Big \rangle \leqslant 0, \ (r=1,2, \cdots , p ), \\&\Big \langle \tilde{\xi }_i^{\Phi }, \eta (x,z) \Big \rangle \leqslant 0, \quad \forall \ i \in \delta \cup \omega , \ \Big \langle \tilde{\xi }_i^{\Psi }, \eta (x,z) \Big \rangle \leqslant 0, \ \forall \ i \in \omega \cup \kappa . \end{aligned}$$

Since \(\omega _\gamma ^{\Phi } \cup \omega _\gamma ^{\Psi } \cup \delta _\gamma ^+ \cup \kappa _\gamma ^+ = \varnothing ,\) we have

$$\begin{aligned}&\Bigg \langle \sum _{i=1 }^m \tau _i^g \tilde{\xi _i^g}, \eta (x,z) \Bigg \rangle \leqslant 0, \quad \Bigg \langle \sum _{r=1}^{p}\left[ \tau _r^h \tilde{\zeta }_r + \gamma _r^h\tilde{\nu }_r \right] , \eta (x,z)\Bigg \rangle \leqslant 0, \\&\Bigg \langle \sum \limits _{\delta \cup \omega } \tau _i^{\Phi } \tilde{\xi }_i^{\Phi }, \eta (x,z) \Bigg \rangle \leqslant 0, \quad \Bigg \langle \sum \limits _{\omega \cup \kappa } \tau _i^{\Psi } \tilde{\xi }_i^{\Psi }, \eta (x,z) \Bigg \rangle \leqslant 0. \end{aligned}$$

Therefore,

$$\begin{aligned} \left\langle \left( \sum _{i=1}^m\tau _i^g \tilde{\xi }_i^g+ \sum _{r=1}^{p}\left[ \tau _r^h\tilde{\zeta }_r+ \gamma _r^h \tilde{\nu }_r \right] + \sum \limits _{\delta \cup \omega }\tau _i^{\Phi } \tilde{\xi }_i^{\Phi } + \sum \limits _{\omega \cup \kappa }\tau _i^{\Psi } \tilde{\xi }_i^{\Psi } \right) , \eta (x,z) \right\rangle \leqslant 0, \end{aligned}$$

which contradicts (4.19). Hence \(F(x) \geqslant F(z)\). This completes the proof.

Theorem 4.6

(Strong Duality) Let \(\tilde{x}\) be a local optimal solution of SIMPPEC and let F be locally Lipschitz near \(\tilde{x}\). Suppose that F is \(\partial ^*\)-pseudoinvex at \(\tilde{x},\) with respect to the kernel \(\eta ,\) further \(g(.,t) \ (t \in T), \ \pm h_r \ (r= 1,2,\cdots , p), \ -\Phi _i \ (i \in \delta \cup \omega ),\ -\Psi _i\ (i \in \omega \cup \kappa )\) admit bounded upper semi-regular convexificators and are \(\partial ^*\)-quasiinvex functions at \(\tilde{x}\) with respect to the common kernel \(\eta .\) If GS-ACQ holds at \( \tilde{x},\) then there exists \(\tilde{\tau }, \) such that \((\tilde{x}, \tilde{\tau })\) is an optimal solution of the MWDP and the respective objective values are equal.

Proof

The proof follows on the lines of the proof of Theorem 4.4 by using Theorem 4.5.

5 Conclusions

We have studied SIMPPEC and established sufficient optimality condition under generalized invexity assumptions. We have introduced Wolfe and Mond–Weir-type dual models for the SIMPPEC in the framework of convexificators. We have established weak and strong duality theorems relating to the SIMPPEC and two dual models using \(\partial ^*\)-invexity, \(\partial ^*\)-pseudoinvexity and \(\partial ^*\)-quasiinvexity assumptions.