Some Results on Mathematical Programs with Equilibrium

Vol.:(0123456789)

Operations Research Forum (2021) 2:53 https://doi.org/10.1007/s43069-021-00061-4

1 3

ORIGINAL RESEARCH

Some Results on Mathematical Programs with Equilibrium Constraints

Bhuwan Chandra Joshi1

Received: 14 February 2020 / Accepted: 4 March 2021 © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2021

AbstractMathematical programs with equilibrium constraints (MPEC) are special class of constrained optimization problems. The feasible set of MPEC violates most of the standard constraint qualifications. Thus, the Karush-Kuhn-Tucker conditions are not necessarily satisfied at minimizers, and the convergence assumptions of many meth-ods for solving constrained optimization problems are not fulfilled. Thus, it is neces-sary, from a theoretical and numerical point of view, to consider suitable optimal-ity conditions for solving such optimization problems. In this paper, we show that M-stationary condition is sufficient for global or local optimality under some mathe-matical programming problem with equilibrium constraints and generalized invexity assumptions. Further, we formulate and study, Wolfe-type and Mond-Weir-type dual models for the MPEC and we establish weak and strong duality theorems relating to the MPEC and the two dual models under invexity and generalized invexity assump-tions. The main purpose of this manuscript is to study the Mathematical programs with equilibrium constraints under the framework of differentiable generalized invex functions and to obtain optimality conditions and duality results.

Keywords Generalized invexity · Wolfe-type dual · Mond-Weir-type dual · Mathematical program with equilibrium constraints

1 Introduction

An MPEC problem is a constraint optimization problem in which the constraints are defined by parametric variational inequalities or complementarity systems. In gen-eral, the MPEC is a highly nonconvex optimization problem and computationally very difficult to solve for a global optimal solution. The MPEC is closely related to the economic problem of Stackelberg game introduced by [1], which is an extension of the renowned Nash game theory given by [2]. Bracken and McGill [3] used the

* Bhuwan Chandra Joshi [email protected]

1 Department of Mathematics, Graphic Era University, Dehradun 248002, India

http://orcid.org/0000-0003-2718-3823

http://crossmark.crossref.org/dialog/?doi=10.1007/s43069-021-00061-4&domain=pdf

Operations Research Forum (2021) 2:53

1 3

53 Page 2 of 18

MPEC in production and marketing decisions with multiple objectives in a competi-tive environment. Pang and Trinkle [4] proposed the model, which is special case of the MPEC, to deal the possible inconsistency of certain motion of robot hands.

Mathematical program with equilibrium constraints (MPEC) includes the bi-level programming problem (see [5, 6]) as its special case. Some feasibility issues for MPEC are presented by Fukushima and Pang [7], in which additional joint con-straints are present and that must be satisfied by the state and design variables of the problem. Scheel and Scholtes [8] introduced several stationary concepts and pre-sented a mathematical program with complementarity constraints(MPCC). In 2011, Henrion and Surowiec [9] compared two distinct calmness conditions on MPEC and derived first-order necessary optimality conditions by using tools of generalized dif-ferentiation introduced by Mordukhovich [10]. Moreover, Gfrerer [11] introduced the concept of strong M-stationarity, which makes a bridge between S-stationarity and M-stationarity for MPEC. For more literature on MPEC, we refer to [12–14] and references therein.

The use of equilibrium constraints in modeling process engineering problems is a relatively new and exciting field of research; see Raghunathan and Biegler [15]. Hydroeconomic river basin models (HERBM) based on mathematical programming are conventionally formulated as explicit aggregate optimization problems with a single, aggregate objective function. Britz et al. [16] proposed a new solution format for hydroeconomic river basin models, based on a multiobjective optimization prob-lem with equilibrium constraints. MPEC has an extensive list of applications such as traffic control, engineering design and economic modeling (see [17, 18]).

In order to solve many practical problems, there have been many attempts to weaken the convexity assumptions. Therefore, a number of concepts on gener-alized convex functions have been introduced and applied to mathematical pro-gramming problems in the literature [19–21]. One of the significant generaliza-tion of convex function is introduced by Hanson [22]. Hanson [22] generalized the Karush-Kuhn-Tucker (KKT)-type sufficient optimality condition with the help of a new class of generalized convex function for differentiable real-valued func-tions which are defined on ℝn . Later, this class of functions was named by Craven [23] as the class of “invex” functions due to its property of invariance under con-vex transformations. The class of invex functions preserves many properties of the class of convex functions and has shown to be very fruitful and useful in a variety of applications [24]. For more literature [25, 26].

For the last three decades, duality and optimality conditions in generalized con-vexity optimization theory have been discussed by several authors [24, 27–29], espe-cially due to the modern work in optimization such as theoretical physics, economic science, mathematical programming, game theory, critical point theory, nonconvex-nonsmooth analysis, variational analysis and in many other areas. In nonlinear pro-gramming problems, Wolfe- and Mond-Weir-type dual models are most popular. The objective function in Mond-Weir dual is the same as that of the primal problem and for generalized convex functions duality results also hold, which is the main advantage of Mond-Weir dual over the Wolfe dual. By using generalized convexity assumptions, Pandey and Mishra [30] and Guu et al. [31] studied Wolfe- and Mond-Weir-type dual models for MPEC and presented weak and strong duality results.

1 3

Operations Research Forum (2021) 2:53 Page 3 of 18 53

The organization of this paper is as follows: in Section 2, we give some prelimi-nary, definitions and results which will be used in the sequel. In Section 3, we show that M-stationary condition is sufficient for global or local optimality under some MPEC generalized invexity assumptions. In Section 4, we formulate Wolfe- and Mond-Weir-type dual models for the MPEC and establish weak and strong duality theorems relating to the MPEC and the two dual models under invexity and general-ized invexity assumptions. In Section 5, we conclude the results of this paper.

2 Preliminaries

In this section, we give some preliminaries and definitions which will be used throughout the paper.

We consider the mathematical program with equilibrium constraints of the form:

where F ∶ ℝn→ ℝ, g ∶ ℝn

→ ℝk, h ∶ ℝn→ ℝp, � ∶ ℝn

→ ℝl and � ∶ ℝn→ ℝl ,

assume that, F, g, h, � and � are continuously differentiable on ℝn.The feasible set of the problem (MPEC) is denoted by X, that is

The following definitions of generalized invex functions are taken from [32].

Definition 1 Let X ⊆ ℝn be an open set. The differentiable function F ∶ X → ℝ is said to be p-invex at v ∈ X if there exist a kernel function � ∶ X × X → ℝn, such that

Definition 2 Let X ⊆ ℝn be an open set. The differentiable function F ∶ X → ℝ is said to be p-pseudoinvex at v ∈ X if there exist a kernel function � ∶ X × X → ℝn, such that

Definition 3 Let X ⊆ ℝn be an open set. The differentiable function F ∶ X → ℝ is said to be p-quasiinvex at v ∈ X if there exist a kernel function � ∶ X × X → ℝn, such that

Given a feasible vector v ∈ X for the problem (MPEC), we define the following index sets:

(MPEC) min F(v)

subject to ∶ g(v) ⩽ 0, h(v) = 0,

�(v) ⩾ 0, �(v) ⩾ 0, ⟨�(v), �(v)⟩ = 0,

X ∶= {v ∈ ℝn ∶ g(v) ⩽ 0, h(v) = 0, �(v) ⩾ 0, �(v) ⩾ 0, ⟨�(v), �(v)⟩ = 0}.

F(v) ⩾ F(v) +1

p⟨∇F(v), ep𝜂(v,v) − �⟩,∀v ∈ X, p ≠ 0.

1

p⟨∇F(v), ep𝜂(v,v) − �⟩ ⩾ 0 ⇒ F(v) ⩾ F(v), ∀v ∈ X, p ≠ 0.

F(v) ⩽ F(v) ⇒1

p⟨∇F(v), ep𝜂(v,v) − �⟩ ⩽ 0,∀v ∈ X, p ≠ 0.


1 3

53 Page 4 of 18

Here the set � is known as degenerate set and if � is empty, the vector v is said to satisfy the strict complementarity condition.

In 1999, Outrata [33] introduced the following concept of M-stationary point.

Definition 4 (M-stationary point) A feasible point v of MPEC is said to be Mordukhovich stationary point if, ∃ � = (�g, �h, ��, ��) ∈ ℝk+p+2l , such that following conditions hold:

Definition 5 (see [34]) (S-stationary point) A feasible point v of MPEC is said to be strong stationary point if, ∃ � = (�g, �h, ��, ��) ∈ ℝk+p+2l , such that the following condition along with Eqs. (1) and (2) holds:

Definition 6 (see [8]) (C-stationary point) A feasible point v of MPEC is said to be Clarke stationary point if, ∃ � = (�g, �h, ��, ��) ∈ ℝk+p+2l , such that the following condition along with Eqs. (1) and (2) holds:

Definition 7 (see [35]) (A-stationary point) A feasible point v of MPEC is said to be alternatively stationary point if, ∃ � = (�g, �h, ��, ��) ∈ ℝk+p+2l , such that the fol-lowing condition along with Eqs. (1) and (2) holds:

Remark 1 Strong stationarity implies M-, A- and C-stationarity and the intersection of A-stationarity and C-stationarity give M-stationarity.

Definition 8 (Definition 2.10 [36]) Let v be a feasible point of MPEC and all func-tions are continuously differentiable at v . We say that the No Nonzero Abnormal Multiplier Constraint Qualification (NNAMCQ) is satisfied at v , if there is no nonzero vector � = (�g, �h, ��, ��) ∈ ℝk+p+2l , such that

Ig ∶=Ig(v) ∶= {i = 1, 2,… , k ∶ gi(v) = 0},

𝛿 ∶=𝛿(v) ∶= {i = 1, 2,… , l ∶ 𝜙i(v) = 0, 𝜃i(v) > 0},

𝜅 ∶=𝜅(v) ∶= {i = 1, 2,… , l ∶ 𝜙i(v) = 0, 𝜃i(v) = 0},

𝛼 ∶=𝛼(v) ∶= {i = 1, 2,… , l ∶ 𝜙i(v) > 0, 𝜃i(v) = 0}.

(1)0 = ∇F(v) +∑i∈Ig

𝛽g

i∇gi(v) +

p∑i=1

𝛽hi∇hi(v) −

l∑i=1

[𝛽𝜙

i∇𝜙i(v) + 𝛽𝜃

i∇𝜃i(v)],

(2)𝛽g

Ig⩾ 0, 𝛽𝜙

𝛼= 0, 𝛽𝜃

𝛿= 0,

∀i ∈ 𝜅, either 𝛽𝜙

i> 0, 𝛽𝜃

i> 0 or 𝛽

𝜙

i𝛽𝜃i= 0.

∀i ∈ �, ��

i⩾ 0, ��

i⩾ 0.

∀i ∈ �, ��

i��i⩾ 0.

∀i ∈ �, ��

i⩾ 0 or ��

i⩾ 0.

1 3


Definition 9 (Definition 2.11 [36]) Let v be a feasible point of MPEC and all functions are continuously differentiable at v. We say that MPEC generalized Mangasarian-Fromovitz constraint qualification (MPEC GMFCQ) is satisfied at v if

(i) for each partition of � into sets P, Q, R with R ≠ � , we have q, such that

and for some i ∈ R either ∇𝜙i(v)Tq > 0 or ∇𝜃i(v)Tq > 0;

(ii) for each partition of � into sets P, Q, the gradient vectors

are linearly independent and there exists q ∈ ℝn, such that

The following proposition demonstrates the relationship between NNAMCQ and MPEC GMFCQ.

Proposition 1 (Proposition 2.1 [36]) NNAMCQ is equivalent to MPEC GMFCQ.

The next theorem gives the Fritz-John-type M-stationary condition for a feasible solution to be a local solution of the MPEC.

Theorem 1 (Theorem 2.1 [36]) (Fritz-John-type M-stationary condition) Let v be a local solution of MPEC, where all functions are continuously differentiable at v . Then there exists r ⩾ 0, � = (�g, �h, ��, ��) ∈ ℝk+p+2l, not all zero, such that

0 =∑i∈Ig

𝛽g

i∇gi(v) +

p∑i=1

𝛽hi∇hi(v) −

l∑i=1

[𝛽𝜙


i∇𝜃i(v)],

𝛽g

Ig⩾ 0, 𝛽𝜙

𝛼= 0, 𝛽𝜃

𝛿= 0,


i> 0, 𝛽𝜃

i> 0 or 𝛽

𝜙

i𝛽𝜃i= 0.

∇gi(v)Tq ⩽ 0, ∀i ∈ Ig,

∇hi(v)Tq = 0, ∀i = 1, 2,… , p,

∇𝜙i(v)Tq = 0, ∀i ∈ 𝛿 ∪ Q,

∇𝜃i(v)Tq = 0, ∀i ∈ 𝛼 ∪ P,

∇𝜙i(v)Tq ⩾ 0, ∇𝜃i(v)

Tq ⩾ 0, i ∈ R,

∇hi(v), ∀ i = 1, 2,… , p,

∇𝜙i(v), ∀i ∈ 𝛿 ∪ Q,

∇𝜃i(v), ∀i ∈ 𝛼 ∪ P,

∇gi(v)Tq < 0, ∀i ∈ Ig,

∇hi(v)Tq = 0, ∀ i = 1, 2,… , p,

∇𝜙i(v)Tq = 0, ∀i ∈ 𝛿 ∪ Q,

∇𝜃i(v)Tq = 0, ∀i ∈ 𝛼 ∪ P.


1 3

53 Page 6 of 18

Suppose r ≠ 0, and let us consider r = 1, then the following Kuhn-Tucker-type M-stationary condition holds.

Corollary 1 (Corollary 2.1 [36]) (Kuhn-Tucker-type necessary M-stationary condition) Let v be a local optimal solution for MPEC where, all the functions are continuously differentiable at v and consider that NNAMCQ is satisfied at v , then v is M-stationary.

In the next section, it can be seen that M-stationary condition turns into a suffi-cient optimality condition under certain MPEC generalized invexity condition.

Note Throughout the paper {} will denote an empty set.

3 Sufficient M‑stationary condition

Theorem 2 Let v be a feasible point of MPEC and M-stationary condition holds at v , i.e., ∃ � = (�g, �h, ��, ��) ∈ ℝk+p+2l , such that

Let

and assume that F is p-pseudoinvex at v, with respect to the kernel �, and a real number p ≠ 0, gi (i ∈ Ig), hi (i ∈ j+), −hi (i ∈ j−), �i (i ∈ �− ∪ �−

�), −�i (i ∈ �+∪

�+

�∪ �+), �

i(i ∈ �− ∪ �−

�) and −�i (i ∈ �+ ∪ �+

�∪ �+) are p-quasiinvex at v, with

0 = r∇F(v) +∑i∈Ig

𝛽g

i∇gi(v) +

p∑i=1

𝛽hi∇hi(v) −

l∑i=1

[𝛽𝜙


i∇𝜃i(v)],

𝛽g

Ig⩾ 0, 𝛽𝜙

𝛼= 0, 𝛽𝜃

𝛿= 0,


i> 0, 𝛽𝜃

i> 0 or 𝛽

𝜙

i𝛽𝜃i= 0.

(3)

0 = ∇F(v) +∑i∈Ig

𝛽g

i∇gi(v) +

p∑i=1

𝛽hi∇hi(v) −

l∑i=1

[𝛽𝜙


i∇𝜃i(v)],

𝛽g

Ig⩾ 0, 𝛽𝜙

𝛼= 0, 𝛽𝜃

𝛿= 0,


i> 0, 𝛽𝜃

i> 0 or 𝛽

𝜙

i𝛽𝜃i= 0.

j+ ∶= {i ∶ 𝛽hi> 0}, j− ∶= {i ∶ 𝛽h

i< 0},

𝜅+ ∶= {i ∈ 𝜅 ∶ 𝛽𝜙

i> 0, 𝛽𝜃

i> 0},

𝜅+

𝜙∶= {i ∈ 𝜅 ∶ 𝛽

𝜙

i= 0, 𝛽𝜃

i> 0}, 𝜅−

𝜙∶= {i ∈ 𝜅 ∶ 𝛽

𝜙

i= 0, 𝛽𝜃

i< 0},

𝜅+

𝜃∶= {i ∈ 𝜅 ∶ 𝛽𝜃

i= 0, 𝛽

𝜙

i> 0}, 𝜅−

𝜃∶= {i ∈ 𝜅 ∶ 𝛽𝜃

i= 0, 𝛽

𝜙

i< 0},

𝛿+ ∶= {i ∈ 𝛿 ∶ 𝛽𝜙

i> 0}, 𝛿− ∶= {i ∈ 𝛿 ∶ 𝛽

𝜙

i< 0},

𝛼+ ∶= {i ∈ 𝛼 ∶ 𝛽𝜃i> 0}, 𝛼− ∶= {i ∈ 𝛼 ∶ 𝛽𝜃

i< 0},

1 3


respect to the common kernel � and for the same real number p ≠ 0. If, 𝛿− ∪ 𝛼− ∪ 𝜅−

𝜙∪ 𝜅−

𝜃= {}, v is a global optimal solution of MPEC; if, �−

�∪ �−

�= {}

or when v is an interior point relative to the set,

i.e., for any feasible point v which is close to v, one has

then v is a local optimal solution of MPEC, where X is the set of feasible solutions of MPEC.

Proof Assume that v be any feasible point of MPEC, i.e., for any i ∈ Ig,

Using p-quasiinvexity of gi at v with respect to the common kernel � , it follows that,

Similarly, we have

Since, for any feasible point v,−�(v) ⩽ 0,−�(v) ⩽ 0, one also have

Case 1 First, we take �− ∪ �− ∪ �−�∪ �−

�= {}, multiplying Eqs. (4)−(8) by

𝛽g

i≥ 0 (i ∈ Ig), 𝛽

hi> 0 (i ∈ j+),−𝛽h

i> 0 (i ∈ j−), 𝛽

𝜙

i> 0 (i ∈ 𝛿+ ∪ 𝜅+

𝜃∪ 𝜅+), 𝛽𝜃

i

> 0 (i ∈ 𝛼+ ∪ 𝜅+

𝜙∪ 𝜅+) , respectively, and adding Eqs. (4)−(8), we obtain

Using Eq. (3), the above inequality follows that

X ∩ {v ∶ �i(v) = 0, �i(v) = 0, i ∈ �−�∪ �−

�},

�i(v) = 0, �i(v) = 0, ∀i ∈ �−�∪ �−

�,

gi(v) ⩽ 0 = gi(v).

(4)1

p⟨∇gi(v), ep𝜂(v,v) − �⟩ ⩽ 0, ∀i ∈ Ig.

(5)1

p⟨∇hi(v), ep𝜂(v,v) − �⟩ ⩽ 0, ∀i ∈ j+,

(6)−1

p⟨∇hi(v), ep𝜂(v,v) − �⟩ ⩽ 0, ∀i ∈ j−.

(7)−1

p⟨∇𝜙i(v), e

p𝜂(v,v) − �⟩ ⩽ 0, ∀i ∈ 𝛿+ ∪ 𝜅+

𝜃∪ 𝜅+,

(8)−1

p⟨∇𝜃i(v), ep𝜂(v,v) − �⟩ ⩽ 0, ∀i ∈ 𝛼+ ∪ 𝜅+

𝜙∪ 𝜅+.

1

p

⟨∑i∈Ig

𝛽g

i∇gi(v) +

p∑i=1

𝛽hi∇hi(v) −

l∑i=1

[𝛽𝜙


i∇𝜃i(v)], e

p𝜂(v,v) − �

⟩⩽ 0.


1 3

53 Page 8 of 18

Applying, p-pseudoinvexity of F at v with respect to the kernel � and for the same real number p ≠ 0 , we get F(v) ⩾ F(v) for all feasible point v. Hence v is a global optimal solution of MPEC.

Case 2 Now, we take �− ∪ �− ≠ {} and �−�∪ �−

�= {}. For v sufficiently close to

v, one has

Applying, p-quasiinvexity of �i(i ∈ �−) at v with respect to the common kernel � , i.e., for v sufficiently close to v , it holds that,

In the same manner, for v sufficiently close to v, one has

Now, multiplying Eqs. (4)−(10) by 𝛽gi⩾ 0 (i ∈ Ig), 𝛽

hi> 0 (i ∈ j+), −𝛽h

i> 0 (i ∈

j−), 𝛽𝜙

i> 0 (i ∈ 𝛿+ ∪ 𝜅+


i> 0 (i ∈ 𝛼+ ∪ 𝜅+

𝜙∪ 𝜅+), −𝛽

𝜙

i> 0 (i ∈ 𝛿−),−𝛽𝜃

i

> 0 (i ∈ 𝛼−) , respectively, and adding, we get

Using Eq. (3), for v sufficiently close to v , the above inequality follows that

Now applying, p-pseudoinvexity of F at v with respect to the kernel � and for the same real number p ≠ 0 , we get F(v) ⩾ F(v) , i.e., v is a local optimal solution of MPEC.

Suppose that v is an interior point relative to the set X ∩ {v ∶ �i(v) = 0,

�i(v) = 0, i ∈ �−

�∪ �−

�}. Then for any feasible point v sufficiently close to v, it holds

that

and hence by p-quasiinvexity of �i (i ∈ �−�) and �i (i ∈ �−

�),

1

p⟨∇F(v), ep𝜂(v,v) − �⟩ ⩾ 0.

𝜙i(v) = 𝜙i(v), ∀i ∈ 𝛿.

(9)1

p⟨∇𝜙i(v), e

p𝜂(v,v) − �)⟩ ⩽ 0, ∀i ∈ 𝛿−.

(10)1

p⟨∇𝜃i(v), ep𝜂(v,v) − �)⟩ ⩽ 0, ∀i ∈ 𝛼−.

1

p

⟨∑i∈Ig

𝛽g

i∇gi(v) +

p∑i=1

𝛽hi∇hi(v) −

l∑i=1

[𝛽𝜙


i∇𝜃i(v)], e

p𝜂(v,v) − �

⟩⩽ 0.

1

p⟨∇F(v), ep𝜂(v,v) − �)⟩ ⩾ 0.

�i(v) = 0, �i(v) = 0, ∀i ∈ �−�∪ �−

�,

(11)1

p⟨∇𝜙i(v), e

p𝜂(v,v) − �)⟩ ⩽ 0, ∀i ∈ 𝜅−𝜃

1 3


Again multiplying Eqs. (4)−(12) by 𝛽gi⩾ 0 (i ∈ Ig), 𝛽

hi> 0 (i ∈ j+), −𝛽h

i> 0 (i ∈

j−), 𝛽𝜙

i> 0 (i ∈ 𝛿+ ∪ 𝜅+


i> 0 (i ∈ 𝛼+ ∪ 𝜅+

𝜙∪ 𝜅+), −𝛽

𝜙

i> 0 (i ∈ 𝛿−),−𝛽𝜃

i

> 0 (i ∈ 𝛼−), −𝛽𝜙

i> 0 (i ∈ 𝛿− ∪ 𝜅−

𝜃), −𝛽𝜃

i> 0 (i ∈ 𝛼− ∪ 𝜅−

𝜙), respectively, and add-

ing, we have

By virtue of Eq. (3), for v sufficiently close to v , the above inequality follows that

By the p-pseudoinvexity of F at v , we have F(v) ⩾ F(v) for v sufficiently close to v. That is, v is a local optimal solution of MPEC and this completes the proof.

4 Duality

In this section, we formulate a Wolfe-type dual problem and a Mond-Weir-type dual problem for the MPEC under p-invexity and generalized p-invexity assumptions.

subject to:

where � = (�g, �h, ��, ��) ∈ ℝk+p+2l.

Theorem 3 (Weak Duality) Let u be feasible for MPEC, (v, �) be feasible for WDMPEC and index sets Ig, �, �, � defined accordingly. Suppose that F, g

i

(i ∈ Ig), hi (i ∈ j+),−hi (i ∈ j−), �i (i ∈ �− ∪ �−�),−�i (i ∈ �+ ∪ �+

�∪ �+), �i (i ∈

�− ∪ �−�), and −�i (i ∈ �+ ∪ �+

�∪ �+) are p-invex functions at v with respect to the

(12)1

p⟨∇𝜃i(v), ep𝜂(v,v) − �)⟩ ⩽ 0, ∀i ∈ 𝜅−

𝜙.

1

p

⟨∑i∈Ig

𝛽g

i∇gi(v) +

p∑i=1

𝛽hi∇hi(v) −

l∑i=1

[𝛽𝜙


i∇𝜃i(v)], e

p𝜂(v,v) − �

⟩⩽ 0.

1

p⟨∇F(v), ep𝜂(v,v) − �⟩ ⩾ 0.

WDMPEC maxv,�

⎧⎪⎨⎪⎩F(v) +

�i∈Ig

�g

igi(v) +

p�i=1

�hihi(v) −

l�i=1

[��

i�i(v) + ��

i�i(v)]

⎫⎪⎬⎪⎭

(13)

0 = ∇F(v) +∑i∈Ig

𝛽g

i∇gi(v) +

p∑i=1

𝛽hi∇hi(v) −

l∑i=1

[𝛽𝜙


i∇𝜃i(v)],

𝛽g

Ig⩾ 0, 𝛽𝜙

𝛼= 0, 𝛽𝜃

𝛿= 0,


i> 0, 𝛽𝜃

i> 0 or 𝛽

𝜙

i𝛽𝜃i= 0,


1 3

53 Page 10 of 18

common kernel � and for the same real number p ≠ 0. If �− ∪ �− ∪ �−�∪ �−

�= {}

then, for any u feasible for the MPEC, we have

Proof Let us consider that, u be any feasible point for MPEC. Then, we get

and

Since, F is p-invex at v, with respect to the kernel � , then

Similarly, we get

If �− ∪ �− ∪ �−�∪ �−

�= {}, multiplying Eqs. (15)−(19) by 𝛽g

i⩾ 0 (i ∈ Ig), 𝛽

hi> 0

(i ∈ j+), −𝛽hi> 0 (i ∈ j−), 𝛽

𝜙

i> 0 (i ∈ 𝛿+ ∪ 𝜅+


i> 0 (i ∈ 𝛼+ ∪ 𝜅+

𝜙∪ 𝜅+),

respectively, and adding Eqs. (14)−(19), it follows that

F(u) ⩾ F(v) +∑i∈Ig

�g

igi(v) +

p∑i=1

�hihi(v) −

l∑i=1

[��

i�i(v) + ��

i�i(v)].

gi(u) ⩽ 0, ∀i ∈ Ig,

hi(u) = 0, i = 1, 2,… , p.

(14)F(u) − F(v) ⩾1

p⟨∇F(v), ep�(u,v) − �⟩.

(15)gi(u) − gi(v) ⩾1

p⟨∇gi(v), ep�(u,v) − �⟩, ∀i ∈ Ig,

(16)hi(u) − hi(v) ⩾1

p⟨∇hi(v), ep�(u,v) − �⟩, ∀i ∈ j+,

(17)−hi(u) + hi(v) ⩾ −1

p⟨∇hi(v), ep�(u,v) − �⟩, ∀i ∈ j−,

(18)−�i(u) + �i(v) ⩾ −1

p⟨∇�i(v), e

p�(u,v) − �⟩, ∀i ∈ �+ ∪ �+

�∪ �+,

(19)−�i(u) + �i(v) ⩾ −1

p⟨∇�i(v), ep�(u,v) − �⟩, ∀i ∈ �+ ∪ �+

�∪ �+.

1 3


Using Eq. (13), we get

Using the feasibility of u for MPEC, that is gi(u) ⩽ 0, hi(u) = 0, �i(u) ⩾ 0,

�i(u) ⩾ 0, we obtain

Hence,

and the proof is complete.

Theorem 4 (Strong Duality) If u is a global optimal solution of MPEC, such that NNAMCQ is satisfied at u and index sets Ig, �, �, � defined accordingly. Let F, gi (i ∈ Ig), hi (i ∈ j+), −hi (i ∈ j−), �i (i ∈ �− ∪ �−

�), −�i (i ∈ �+ ∪ �+

�∪ �+),

�i(i ∈ �− ∪ �−

�) and −�i (i ∈ �+ ∪ �+

�∪ �+) fulfill the assumption of Theorem 3.

Then, there exists 𝛽, such that (u, 𝛽) is a global optimal solution of WDMPEC and corresponding objective values of MPEC and WDMPEC are equal.

Proof Since u is a global optimal solution of MPEC and NNAMCQ is satisfied at u , there exist 𝛽 = (𝛽g, 𝛽h, 𝛽𝜙, 𝛽𝜃) ∈ ℝk+p+2l, such that M- stationarity conditions are satisfied for MPEC, i.e.,

F(u) − F(v) +∑i∈Ig

�g

igi(u) −

∑i∈Ig

�g

igi(v) +

p∑i=1

�hihi(u) −

p∑i=1

�hihi(v)

−

l∑i=1

��

i�i(u) +

l∑i=1

��

i�i(v) −

l∑i=1

��i�i(u) +

l∑i=1

��i�i(v)

⩾1

p

⟨∇F(v) +

∑i∈Ig

�g

i∇gi(v) +

p∑i=1

�hi∇hi(v) −

l∑i=1

[��

i∇�i(v) + ��

i∇�i(v)], e

p�(u,v) − �

⟩.

F(u) − F(v) +∑i∈Ig

�g

igi(u) −

∑i∈Ig

�g

igi(v) +

p∑i=1

�hihi(u)

−

p∑i=1

�hihi(v)

l∑i=1

��

i�i(u) +

l∑i=1

��

i�i(v)

−

l∑i=1

��i�i(u) +

l∑i=1

��i�i(v) ⩾ 0.

F(u) − F(v) −∑i∈Ig

�g

igi(v) −

p∑i=1

�hihi(v) +

l∑i=1

[��

i�i(v) + ��

i�i(v)] ⩾ 0.

F(u) ⩾ F(v) +∑i∈Ig

�g

igi(v) +

p∑i=1

�hihi(v) −

l∑i=1

[��

i�i(v) + ��

i�i(v)],


1 3

53 Page 12 of 18

Therefore, (u, 𝛽) is feasible for WDMPEC. By Theorem 3, we get

for any feasible solution (v, �) for WDMPEC. Now, using the feasibility condi-tion of MPEC and WDMPEC, i.e., for i ∈ Ig(u), gi(u) = 0, also h

i(u) = 0,𝜙

i(u)

= 0,∀i ∈ � ∪ � and 𝜃i(u) = 0,∀i ∈ 𝜅 ∪ 𝛼, then, it follows that

Using Eqs. (21) and (22), we get

Therefore, (u, 𝛽) is a global optimal solution for WDMPEC. Moreover, the cor-responding objective values of MPEC and WDMPEC are equal.

Now, we establish the duality relation between the MPEC and the following Mond-Weir-type dual.

subject to:

(20)

0 = ∇F(u) +∑i∈Ig

𝛽g

i∇gi(u) +

p∑i=1

𝛽hi∇hi(u) −

l∑i=1

[𝛽𝜙

i∇𝜙i(u) + 𝛽𝜃

i∇𝜃i(u)],

𝛽g

Ig⩾ 0, 𝛽𝜙

𝛼= 0, 𝛽𝜃

𝛿= 0,


i> 0, 𝛽𝜃

i> 0, or 𝛽

𝜙

i𝛽𝜃i= 0.

(21)F(u) ⩾ F(v) +∑i∈Ig

𝛽g

igi(v) +

p∑i=1

𝛽hihi(v) −

l∑i=1

[𝛽𝜙

i𝜙i(v) + 𝛽𝜃

i𝜃i(v)],

(22)F(u) = F(u) +∑i∈Ig

𝛽g

igi(u) +

p∑i=1

𝛽hihi(u) −

l∑i=1

[𝛽𝜙

i𝜙i(u) + 𝛽𝜃

i𝜃i(u)].

F(u) +∑i∈Ig

𝛽g

igi(u) +

p∑i=1

𝛽hihi(u) −

l∑i=1

[𝛽𝜙

i𝜙i(u) + 𝛽𝜃

i𝜃i(u)]

⩾ F(v) +∑i∈Ig

𝛽g

igi(v) +

p∑i=1

𝛽hihi(v) −

l∑i=1

[𝛽𝜙

i𝜙i(v) + 𝛽𝜃

i𝜃i(v)].

MWDMPEC maxv,�

F(v)

1 3


where � = (�g, �h, ��, ��) ∈ ℝk+p+2l.

Theorem 5 (Weak Duality) Let u be feasible for MPEC, (v, �) be feasible for MWDMPEC and the index sets Ig, �, �, � defined accordingly. Suppose that F, gi (i ∈ Ig), hi (i ∈ j+),−hi (i ∈ j−), �i (i ∈ �− ∪ �−

�), −�i (i ∈ �+ ∪ �+

�∪ �+),

�i(i ∈ �− ∪ �−

�), −�

i(i ∈ �+ ∪ �+

�∪ �+) are p-invex functions at v with respect to

the common kernel � and for the same real number p ≠ 0. If �− ∪ �− ∪ �−�∪ �−

�= {},

then, for any u feasible for the MPEC, we have

Proof Let us consider that, u be any feasible point for MPEC. Then, we have

and

Since, F is p-invex at v with respect to the kernel � , we have

Similarly, we have

(23)

0 = ∇F(v) +∑i∈Ig

𝛽g

i∇gi(v) +

p∑i=1

𝛽hi∇hi(v) −

l∑i=1

[𝛽𝜙


i∇𝜃i(v)],

∑i∈Ig

𝛽g

igi(v) ⩾ 0,

p∑i=1

𝛽hihi(v) ⩾ 0,

l∑i=1

𝛽𝜙

i𝜙i(v) ⩽ 0,

l∑i=1

𝛽𝜃i𝜃i(v) ⩽ 0,

𝛽g

Ig⩾ 0, 𝛽𝜙

𝛼= 0, 𝛽𝜃

𝛿= 0,


i> 0, 𝛽𝜃

i> 0 or 𝛽

𝜙

i𝛽𝜃i= 0,

F(u) ⩾ F(v).

gi(u) ⩽ 0, ∀i ∈ Ig,

hi(u) = 0, i = 1, 2,… , p.

(24)F(u) − F(v) ⩾1

p⟨∇F(v), ep�(u,v) − �⟩.

(25)gi(u) − gi(v) ⩾1

p⟨∇gi(v), ep�(u,v) − �⟩, ∀i ∈ Ig,

(26)hi(u) − hi(v) ⩾1

p⟨∇hi(v), ep�(u,v) − �⟩, ∀i ∈ j+,

(27)−hi(u) + hi(v) ⩾ −1

p⟨∇hi(v), ep�(u,v) − �⟩, ∀i ∈ j−,


1 3

53 Page 14 of 18

If �− ∪ �− ∪ �−�∪ �−

�= {}, multiplying Eqs. (25)−(29) by 𝛽g

i⩾ 0 (i ∈ Ig), 𝛽

hi> 0

(i ∈ j+), −𝛽hi> 0 (i ∈ j−), 𝛽

𝜙

i> 0 (i ∈ 𝛿+ ∪ 𝜅+


i> 0 (i ∈ 𝛼+ ∪ 𝜅+

𝜙∪ 𝜅+),

respectively, and adding Eqs. (24)−(29), we obtain

Using Eq. (23), it follows that

Using the feasibility of u and v for MPEC and MWDMPEC, respectively, we obtain

and the proof is complete.

Theorem 6 (Strong Duality) If u is a global optimal solution of MPEC, such that the NNAMCQ is satisfied at u and index sets Ig, �, �, � defined accordingly. Let F, gi (i ∈ Ig), hi (i ∈ j+), −hi (i ∈ j−), �i (i ∈ �− ∪ �−

�), −�i (i ∈ �+ ∪ �+

�∪ �+),

�i(i ∈ �− ∪ �−

�) and −�i (i ∈ �+ ∪ �+


Then, there exists 𝛽, such that (u, 𝛽) is a global optimal solution of MWDMPEC and corresponding objective values of MPEC and MWDMPEC are equal.

Proof The proof is similar to the proof of Theorem 4.

Now, using generalized p-invexity assumptions, we establish weak and strong duality theorems for the MPEC and its Mond-Weir-type dual problem.

(28)−�i(u) + �i(v) ⩾ −1

p⟨∇�i(v), e

p�(u,v) − �⟩, ∀i ∈ �+ ∪ �+

�∪ �+,

(29)−�i(u) + �i(v) ⩾ −1

p⟨∇�i(v), ep�(u,v) − �⟩, ∀i ∈ �+ ∪ �+

�∪ �+.

F(u) − F(v) +∑i∈Ig

�g

igi(u) −

∑i∈Ig

�g

igi(v) +

p∑i=1

�hihi(u) −

p∑i=1

�hihi(v)

−

l∑i=1

��

i�i(u) +

l∑i=1

��

i�i(v) −

l∑i=1

��i�i(u) +

l∑i=1

��i�i(v)

⩾1

p

⟨∇F(v) +

∑i∈Ig

�g

i∇gi(v) +

p∑i=1

�hi∇hi(v) −

l∑i=1

[��

i∇�i(v) + ��

i∇�i(v)], e

p�(u,v) − �

⟩.

F(u) − F(v) +∑i∈Ig

�g

igi(u) −

∑i∈Ig

�g

igi(v) +

p∑i=1

�hihi(u) −

p∑i=1

�hihi(v)

−

l∑i=1

��

i�i(u) +

l∑i=1

��

i�i(v) −

l∑i=1

��i�i(u) +

l∑i=1

��i�i(v) ⩾ 0.

F(u) ⩾ F(v),

1 3


Theorem 7 (Weak Duality) Let u be feasible for the MPEC, (v, �) be feasible for MWDMPEC and index sets Ig, �, �, � defined accordingly. Also, suppose that F is p-pseudoinvex at v with respect to the kernel � and the functions

and

are p-quasiinvex at v with respect to the common kernel � and for the same real number p ≠ 0 . If �− ∪ �− ∪ �−

�∪ �−

�= {}, then, for any u feasible for the MPEC, we

have

Proof By feasibility conditions of MPEC, MWDMPEC and p-quasiinvexity of ∑i∈Ig

�g

igi(.), we obtain

Similarly, by p-quasiinvexity of ∑p

i=1�hihi(.), we have

For i ∈ �+ ∪ �+

�∪ �+, we get

Similarly, for i ∈ �+ ∪ �+

�∪ �+, we have

Adding Eqs. (30)−(33), we obtain

∑i∈Ig

𝛽g

igi(.)(𝛽

g

i⩾ 0),

p∑i=1

𝛽hihi(.)(𝛽

hi> 0,∀i ∈ j+, 𝛽h

i< 0,∀i ∈ j−),

∑i∈𝛿+∪𝜅+

𝜃∪𝜅+

𝛽𝜙

i(−𝜙i)(.)(𝛽

𝜙

i> 0),

∑i∈𝛿+∪𝜅+

𝜙∪𝜅+

𝛽𝜃i(−𝜃i)(.)(𝛽

𝜃

i> 0),

F(u) ⩾ F(v).

(30)∑i∈Ig

�g

igi(u) −

∑i∈Ig

�g

igi(v) ⩽ 0 ⇒

1

p

⟨∑i∈Ig

�g

i∇gi(v), e

p�(u,v) − �

⟩⩽ 0.

(31)p∑i=1

�hihi(u) −

p∑i=1

�hihi(v) ⩽ 0 ⇒

1

p

⟨ p∑i=1

�hi∇hi(v), e

p�(u,v) − �

⟩⩽ 0.

(32)

l∑i=1

��

i(−�i)(u) −

l∑i=1

��

i(−�i)(v) ⩽ 0 ⇒

1

p

⟨ l∑i=1

��

i(−∇�i)(v), e

p�(u,v) − �

⟩⩽ 0.

(33)

l∑i=1

��i(−�i)(u) −

l∑i=1

��i(−�i)(v) ⩽ 0 ⇒

1

p

⟨ l∑i=1

��i(−∇�i)(v), e

p�(u,v) − �

⟩⩽ 0.


1 3

53 Page 16 of 18

Now, using Eq. (23) and p-pseudoinvexity of F at v with respect to kernel �, it follows that

This completes the proof.

Theorem 8 (Strong Duality) If u is a global optimal solution of MPEC, such that the NNAMCQ is satisfied at u and index sets Ig, �, �, � defined accordingly. Let F, gi (i ∈ Ig), hi (i ∈ j+), −hi (i ∈ j−), �i (i ∈ �− ∪ �−

�), −�i (i ∈ �+ ∪ �+

�∪ �+),

�i(i ∈ �− ∪ �−

�) and −�i (i ∈ �+ ∪ �+


Then, there exists 𝛽, such that (u, 𝛽) is a global optimal solution of MWDMPEC and corresponding objective values of MPEC and MWDMPEC are equal.

Proof Since u is a global optimal solution of MPEC and NNAMCQ is satisfied at u , i.e., there exist 𝛽 = (𝛽g, 𝛽h, 𝛽𝜙, 𝛽𝜃) ∈ ℝk+p+2l, such that the M-stationarity condi-tions are satisfied for MPEC, i.e.,

Since u is an optimal solution for MPEC, it follows that

Therefore, (u, 𝛽) is feasible for MWDMPEC. Now, using Theorem 7, i.e., for any feasible solution (v, �), we obtain

Hence, (u, 𝛽) is an optimal solution for MWDMPEC. Moreover, corresponding objective values of MPEC and MWDMPEC are equal. This completes the proof.

1

p

⟨∑i∈Ig

�g

i∇gi(v) +

p∑i=1

�hi∇hi(v) − [

l∑i

��

i(∇�i)(v) +

l∑i

��i(∇�i)(v)], e

p�(u,v) − �

⟩⩽ 0.

1

p⟨∇F(v), ep�(u,v) − �⟩ ⩾ 0 ⇒ F(u) ⩾ F(v).

0 = ∇F(u) +∑i∈Ig

𝛽g

i∇gi(u) +

p∑i=1

𝛽hi∇hi(u) −

l∑i=1

[𝛽𝜙

i∇𝜙i(u) + 𝛽𝜃

i∇𝜃i(u)],

𝛽g

Ig⩾ 0, 𝛽𝜙

𝛼= 0, 𝛽𝜃

𝛿= 0,∀i ∈ 𝜅, either 𝛽

𝜙

i> 0, 𝛽𝜃

i> 0, or 𝛽

𝜙

i𝛽𝜃i= 0.

∑i∈Ig

𝛽g

igi(u) = 0,

p∑i=1

𝛽hihi(u) = 0,

l∑i=1

𝛽𝜙

i𝜙i(u) = 0,

l∑i=1

𝛽𝜃i𝜃i(u) = 0.

F(u) ⩾ F(v).

1 3


5 Conclusions

We have shown that M-stationary condition is sufficient for global or local optimal-ity under some MPEC generalized p-invexity assumptions. We have also formu-lated the Wolfe-type and Mond-Weir-type dual models for the MPEC. Further, we established weak and strong duality theorems relating the MPEC and the two dual models.

Further, the results of the paper may be extended to the nonsmooth case as well as to more general class of functions (see, [24]). We can also extend the results of this paper to the Lagrange-type dual models. Because, Lagrange-type dual model for the MPEC may be easier to deal from algorithmic point of view rather than other known MPEC dual models, since the dual objective function is concave and the complementarity constraints are not part of it. However, the usefulness of the Lagrange duality for the MPEC depends greatly upon how easy it is to solve the minimization of the MPEC Lagrangian.

Acknowledgements The author is thankful to the anonymous reviewers for their insightful comments and suggestions.

Funding This research received no specific grant from any funding agency in the public, commercial or not for profit sectors.

Data Availability This research article contains all the data generated and materials obtained during this study.

Declarations

Competing Interest The author declares that he has no competing interests.

References

1. Stackelberg HV (1952) The Theory of Market Economy. Oxford University, Press, Oxford 2. Nash JF (1951) Non-cooperative Games. Anal Math 54:286–295 3. Bracken J, McGill JT (1978) Production and marketing decisions with multiple objectives in a com-

petitive environment. J Optim Theory Appl 24(3):449–458 4. Pang JS, Trinkle JC (1996) Complementarity formulations and existence of solutions of dynamic

multi-rigid-body contact problems with Coulomb friction. Math Prog 73(2):199–226 5. Falk JE, Liu J (1995) On bilevel programming, part I: general nonlinear cases. Math Program

70(1–3):47–72 6. Vicente LN, Calamai P (1994) Bilevel and multilevel programming: A bibliography review. J

Global Optim 5(3):291–306 7. Fukushima M, Pang JS (1998) Some feasibility issues in mathematical programs with equilibrium

constraints. SIAM J Optim 8(3):673–681 8. Scheel H, Scholtes S (2000) Mathematical programs with complementarity constraints: Stationarity,

optimality, and sensitivity. Math Oper Res 25(1):1–22 9. Henrion R, Surowiec T (2011) On calmness conditions in convex bilevel programming. Appl Anal

90(6):951–970 10. Mordukhovich BS (2006) Variational analysis and generalized differentiation I: Basic The-

ory. Grundlehren der Mathematischen Wissenschaften [FundamentalPrinciples of Mathematical Sciences], vol 330. Springer-Verlag, Berlin, pp xxii+579


1 3

53 Page 18 of 18

11. Gfrerer H (2014) Optimality conditions for disjunctive programs based on generalized differen-tiation with application to mathematical programs with equilibrium constraints. SIAM J. Optim. 24(2):898–931

12. Joshi BC, Mishra SK (2019) On nonsmooth mathematical programs with equilibrium constraints using generalized convexity. Yugosl J Oper Res 29(4):449–463

13. Joshi BC, Mishra SK, Pankaj (2019) On Semi-infinite Mathematical Programming Problems with Equilibrium Constraints Using Generalized Convexity. J Oper Res Soc China 8(4):619–636

14. Joshi BC (2020) Optimality and duality for nonsmooth semi-infinite mathematical program with equilibrium constraints involving generalized invexity of order 𝜎 > 0 . RAIRO Oper Res. https:// doi. org/ 10. 1051/ ro/ 20200 81

15. Raghunathan AU, Biegler LT (2003) Mathematical programs with equilibrium constraints (MPECs) in process engineering. Comput Chem Eng 27(10):1381–1392

16. Britz W, Ferris M, Kuhn A (2013) Modeling water allocating institutions based on multiple optimi-zation problems with equilibrium constraints. Environ Model Softw 46:196–207

17. Anandalingam G, Friesz TL (eds) (1992) Hierarchical Optimization: An introduction. Ann Oper Res 34(1):1–11

18. Harker PT, Pang JS (1990) Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications. Math Program 48(1–3):161–220

19. Joshi BC, Pankaj Mishra SK (2020) On nonlinear complementarity problems with applications. J Inf Opt Sci. https:// doi. org/ 10. 1080/ 02522 667. 2020. 17373 80

20. Joshi BC (2020) On generalized approximate convex functions and variational inequalities. RAIRO Oper Res. https:// doi. org/ 10. 1051/ ro/ 20201 41

21. Pini R, Singh C (1997) A survey of recent [1985-1995] advances in generalized convexity with applications to duality theory and optimality conditions. Optimization 39(4):311–360

22. Hanson MA (1981) On sufficiency of the Kuhn-Tucker conditions. J Math Anal Appl 80(2):545–550 23. Craven BD (1981) Invex function and constrained local minima. Bull Aust Math Soc 24(3):357–366 24. Mishra SK, Giorgi G (2008) Invexity and Optimization. Springer-Verlag, Heidelberg 25. Giannessi F, Maugeri A, Pardalos PM. (Eds.) (2006) Equilibrium problems: nonsmooth optimiza-

tion and variational inequality models (Vol. 58). Springer Science & Business Media 26. Daniele P, Giannessi F, Maugeri A (eds) (2003) Equilibrium problems and variational models (Vol.

68). Kluwer Academic Publishers, Dordrecht 27. Ben-Israel A, Mond B (1986) What is Invexity. J Aust Math Soc Ser B 28(1):1–9 28. Mishra SK (1997) Second order generalized invexity and duality in mathematical programming.

Optimization 42(1):51–69 29. Mishra SK, Rueda NG (2002) Higher-order generalized invexity and duality in nondifferentiable

mathematical programming. J Math Anal Appl 272(2):496–506 30. Pandey Y, Mishra SK (2016) Duality for nonsmooth optimization problems with equilibrium con-

straints, using convexificators. J Optim Theory Appl 171(2):694–707 31. Guu S-M, Mishra SK, Pandey Y (2016) Duality for nonsmooth mathematical programming prob-

lems with equilibrium constraints. J Inequal Appl 28(1):1–15 32. Antczak T (2001) On (p, r)-invexity-type nonlinear programming problems. J Math Anal Appl

264(2):382–397 33. Outrata JV (1999) Optimality conditions for a class of mathematical programs with equilibrium

constraints. Math Oper Res 24(3):627–644 34. Pang JS, Fukushima M (1999) Complementarity constraint qualifications and simplified B-stationarity

conditions for mathematical programs with equilibrium constraints. Comput Optim Appl 13(1–3):111–136 35. Flegel ML, Kanzow C (2003) A Fritz John approach to first order optimality conditions for math-

ematical programs with equilibrium constraints. Optimization 52(3):277–286 36. Ye JJ (2005) Necessary and sufficient optimality conditions for mathematical programs with equi-

librium constraints. J Math Anal Appl 307(1):350–369

Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

https://doi.org/10.1051/ro/2020081

https://doi.org/10.1051/ro/2020081

https://doi.org/10.1080/02522667.2020.1737380

https://doi.org/10.1051/ro/2020141

Documents

Some Results on Mathematical Programs with Equilibrium