Minimax theorems for scalar set-valued mappings with nonconvex domains and applications

Zhang, Y.; Li, S. J.

doi:10.1007/s10898-012-9992-2

Cited by 12 publications

(5 citation statements)

References 31 publications

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“…Then, the sets

⋃_{y \in Y} max ⋃_{x \in X} F (x, y)

and

⋃_{x \in X} min ⋃_{y \in Y} F (x, y)

are compact, and hence have a minimum and a maximum. Therefore, in this case, if the minimax equality holds for F , then it can be rewritten as

min ⋃_{y \in Y} max ⋃_{x \in X} F (x, y) = max ⋃_{x \in X} min ⋃_{y \in Y} F (x, y) .

To the best of our knowledge, an equality of type (6.1) has not yet been established for set‐valued bifunctions, while sufficient conditions for (6.2) to hold were given by several authors , , . In this section, based on results of Section , we give necessary and sufficient conditions for (8) to hold.…”

Section: Minimax Theorems For Set‐valued Bifunctionsmentioning

confidence: 95%

“…In , , a point

(\bar{x}, \bar{y}) \in X \times Y

is called a saddle point of F iff,

\begin{matrix} true \\ right trueprefixmax ⋃_{x \in X} F (x, \bar{y}) = F (\bar{x}, \bar{y}) = trueprefixmin ⋃_{y \in Y} F (\bar{x}, y) . \end{matrix}

Note that we always have

F^{*} \geq F_{*}

. Hence, if F has a saddle point, then

inf ⋃_{y \in Y} max ⋃_{x \in X} F (x, y) = sup ⋃_{x \in X} min ⋃_{y \in Y} F (x, y) .

It is not hard to see that

(\bar{x}, \bar{y})

is a saddle point of F if and only if

(\bar{x}, \bar{y})

is a solution of problem (VR) with both

A, X

being

X \times Y

S ((x, y)) \equiv X \times Y

, and

R ((x, y), (a, b)) holding iff sup F (a, y) \leq inf F (x, b) .

Moreover, (6.1) holds if and only if, for any real number

…”

Section: Minimax Theorems For Set‐valued Bifunctionsmentioning

confidence: 99%

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Topologically-based characterizations of the existence of solutions of optimization-related problems

Khánh

Quan

2015

Math. Nachr.

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Necessary and sufficient conditions for the existence of solutions of optimization‐related problems, defined on general sets, are established by using topologically‐based structures, without the usual linear structure. First, we prove that the existence of solutions to a variational relation problem is equivalent to the existence of either a KKM‐structure or a connectedness structure, satisfying additionally some verifiable conditions. Then, applying these results, we obtain such full (two‐way) characterizations for the existence of invariant points, solutions of quasiequilibrium problems of the Stampacchia‐Minty type, saddle points, and Nash equilibria for noncooperative games. The main advantages of our scheme with the mentioned two structures are that full characterizations for existence are obtained in a unified way, with no convexity assumptions, and also that one has flexibility to choose a suitable structure in investigations. (This flexibility is decisive when the natural underlying structure of the given problem is scarcely employed.)

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“…Then, the sets

⋃_{y \in Y} max ⋃_{x \in X} F (x, y)

and

⋃_{x \in X} min ⋃_{y \in Y} F (x, y)

are compact, and hence have a minimum and a maximum. Therefore, in this case, if the minimax equality holds for F , then it can be rewritten as

min ⋃_{y \in Y} max ⋃_{x \in X} F (x, y) = max ⋃_{x \in X} min ⋃_{y \in Y} F (x, y) .

Section: Minimax Theorems For Set‐valued Bifunctionsmentioning

confidence: 95%

“…In , , a point

(\bar{x}, \bar{y}) \in X \times Y

is called a saddle point of F iff,

\begin{matrix} true \\ right trueprefixmax ⋃_{x \in X} F (x, \bar{y}) = F (\bar{x}, \bar{y}) = trueprefixmin ⋃_{y \in Y} F (\bar{x}, y) . \end{matrix}

Note that we always have

F^{*} \geq F_{*}

. Hence, if F has a saddle point, then

inf ⋃_{y \in Y} max ⋃_{x \in X} F (x, y) = sup ⋃_{x \in X} min ⋃_{y \in Y} F (x, y) .

It is not hard to see that

(\bar{x}, \bar{y})

is a saddle point of F if and only if

(\bar{x}, \bar{y})

is a solution of problem (VR) with both

A, X

being

X \times Y

S ((x, y)) \equiv X \times Y

, and

R ((x, y), (a, b)) holding iff sup F (a, y) \leq inf F (x, b) .

Moreover, (6.1) holds if and only if, for any real number

…”

Section: Minimax Theorems For Set‐valued Bifunctionsmentioning

confidence: 99%

Topologically-based characterizations of the existence of solutions of optimization-related problems

Khánh

Quan

2015

Math. Nachr.

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show abstract

“…To get the concept of saddle point for a general multi-valued mapping we refer to [4,22] and expand the corresponding concepts in the following definitions. Definition 4.1.…”

Section: Continuing This Process We Get Thatmentioning

confidence: 99%

On equilibrium multivalued problems in the general model with the Boolean-valued bifunction

Tri

2023

NACO

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In this paper, we establish some sufficient conditions for the existence of a solution of the equilibrium problem. The problem is understood in general form: find x ∈ E such that Φ(x, y) (−D\{0}) c for all y ∈ E; and some of its variants, where is a Boolean-valued bifunction and Φ is a multivalued mapping with values in vector space without topological structure. We use this result to show that (C, )-saddle point exists for multi-valued mappings. The results are established by combining the concepts of cyclic quasi-monotonicity and "algebraic" semicontinuity. Our results are interesting and refreshing because we do not need to use the convex hypothesis.1. Introduction. Let E and F be two non-empty sets and given a bifunction Φ : E × F → R. The problem which consists of finding an element x ∈ E satisfying Φ(x, y) ≥ 0 for all y ∈ F(1)has been studied previously in the works of Ky Fan [6], Muu and Oettli [18], Blum and Oettli [2] and others, with the term "equilibrium problem" (in short, EP Φ ). This problem and its related topics play important roles in many different areas of sciences, such as electrical circuits, economics, finance, game theory, constrained optimization, traffic network equilibrium problem, etc (see, e.g., [1,5,15,17,[19][20][21]).The variations of the problem (EP Φ ) are often considered with convex assumptions in a topologically structured environment, such as, in [15], the authors study the second-order optimality conditions for a class of the circular conic optimization problem, in that, the explicit expressions of the tangent cone and the second-order tangent set are established to discuss the minimal problems. In the work [16], the authors study the solubilities of optimization problems associated with the secondorder cone.The study of EP Φ has been extended from scalar to the vector case (for instance, [3,20] and the references therein). To formulate the problem (1), let V be a real

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“…Recently, Zhang and Chen et al [3] studied existence of general n person noncooperative game problems and minimax regret equilibria with set payoff by using Kakutani-Fan-Glicksberg fixed point theorem and a nonlinear scalarization function. For more information, refer to [4] [5] [6] [7].…”

Section: Introductionmentioning

confidence: 99%

Stability of Generalized Minimax Regret Equilibria with Scalar Set Payoff

Zhao¹

2022

JAMP

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In this paper, we first introduce the notion and model of generalized minimax regret equilibria with scalar set payoffs. After that, we study its general stability theorem under the conditions that the existence theorem of generalized minimax regret equilibrium point with scalar set payoffs holds. In other words, when the scalar set payoffs functions and feasible constraint mappings are slightly disturbed, by using Fort theorem and continuity results of set-valued mapping optimal value functions, we obtain a general stability theorem for generalized minimax regret equilibria with scalar set payoffs. At the same time, an example is given to illustrate our result.

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Minimax theorems for scalar set-valued mappings with nonconvex domains and applications

Cited by 12 publications

References 31 publications

Topologically-based characterizations of the existence of solutions of optimization-related problems

Topologically-based characterizations of the existence of solutions of optimization-related problems

On equilibrium multivalued problems in the general model with the Boolean-valued bifunction

Stability of Generalized Minimax Regret Equilibria with Scalar Set Payoff

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