This paper is concerned with solving a stochastic variational inequality problem (for short, SVIP) from a viewpoint of minimization of mixed conditional value-at-risk (CVaR). The regularized gap function for SVIP is used to define a loss function for the SVIP and mixed CVaR to measure the loss. In this setting, SVIP can be reformulated as a deterministic minimization problem. We show that the reformulation is a convex program for a huge class of SVIP under suitable conditions. Since mixed CVaR involves the plus function and mathematical expectation, the neural network smoothing function and Monte Carlo method are employed to get an approximation problem of the minimization reformulation. Finally, we consider the convergence of optimal solutions and stationary points of the approximation.
In this paper, some characterizations for the solution sets of a class of set-valued vector
mixed variational inequalities to be nonempty and bounded are
presented in real reflexive Banach spaces. An equivalence relation between
the solution sets of the vector mixed variational inequalities and the
weakly efficient solution sets of the vector optimization problems is
shown under some suitable assumptions. By using some known results for
the vector optimization problems, several characterizations for the
solution sets of the vector mixed variational inequalities are obtained
in real reflexive Banach spaces. Furthermore, some stability results
for the vector mixed variational inequality are given when the
mapping and the constraint set are perturbed by two different parameters. Finally, the upper semicontinuity and the lower semicontinuity of the
solution sets are given under some suitable assumptions which are different from the ones used in [7, 11, 22]. Some examples are also given to illustrate our results.
The purpose of this paper is to investigate the
nonemptiness and boundedness of solution set for a generalized mixed
variational inequality problem with strict feasibility in reflexive
Banach spaces. We introduce a concept of strict feasibility for the
generalized mixed variational inequality problem which includes the
existing concepts of strict feasibility introduced for variational
inequalities and complementarity problems. By using a degree theory
developed in Wang and Huang , we prove that the monotone
generalized mixed variational inequality has a nonempty bounded
solution set if and only if it is strictly feasible. The results
presented in this paper generalize and extend some known results in
In this paper, a scalarization result and a density theorem concerned with the sets of weakly efficient and efficient approximate solutions to a generalized vector equilibrium problem are given, respectively. By using the scalarization result and the density theorem, the connectedness of the sets of weakly efficient and efficient approximate solutions to the generalized vector equilibrium problem are established without the assumptions of monotonicity and compactness. The lower semicontinuity of weakly efficient and efficient approximate solution mappings to the parametric generalized vector equilibrium problem with perturbing both the objective mapping and the feasible region are obtained without the assumptions of monotonicity and compactness. Furthermore, the upper semicontinuity of weakly efficient approximate solution mapping and the Hausdorff upper semicontinuity of efficient approximate solution mapping to the parametric generalized vector equilibrium problem with perturbing both the objective mapping and the feasible region are also given under some suitable conditions.
In this paper, we consider a Wishart Affine Stochastic Correlation (WASC) model which accounts for the stochastic volatilities of the assets and for the stochastic correlations not only between the underlying assets' returns but also between their volatilities. Under the assumptions of the model, we derive the neutral and indifference pricing for general European-style financial contracts. The paper shows that comparing to risk-neutral pricing, the utility-based pricing methods are generally feasible and avoid factitiously dealing with some risk premia corresponding to the volatilities-correlations as a consequence of the incompleteness of the market.
In this paper, we introduce the concept of
well-posedness for the vector quasi-equilibrium problem. We obtain
some necessary and sufficient conditions for well-posedness of
vector quasi-equilibrium problems. As applications, we investigate
the well-posedness for vector quasi-variational inequality problems
and vector quasi-optimization problems.
In this paper, a class of
nondifferentiable multiobjective fractional programs is
studied, in which every component of the objective function contains a term involving the
support function of a compact convex set. Kuhn-Tucker necessary
and sufficient optimality conditions, duality and saddle point results for weakly efficient
solutions of the nondifferentiable multiobjective fractional programming problems are given.
The results presented in this paper improve and extend some the corresponding results in the
In this paper, we study the Levitin-Polyak type
well-posedness of variational inequalities and optimization problems
with variational inequality constraints in Banach spaces. We derive
some criteria and characterizations for these Levitin-Polyak
well-posedness. We also investigate conditions under which the
existence and uniqueness of solution is equivalent to the
Levitin-Polyak well-posedness of the problem.
This Special Issue of Numerical Algebra, Control and Optimization (NACO) is dedicated to Professor Franco Giannessi on the occasion of his 75th birthday and in recognition of his many fundamental contributions in Optimization and Nonlinear Analysis. It is a great honor and pleasure for the Guest Editors to have this opportunity to edit this Special Issue.
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