JIMO
Neural network smoothing approximation method for stochastic variational inequality problems
Hui-Qiang Ma Nan-Jing Huang
Journal of Industrial & Management Optimization 2015, 11(2): 645-660 doi: 10.3934/jimo.2015.11.645
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.
keywords: regularized gap function neural network smoothing approximation Stochastic variational inequality Monte Carlo sampling approximation convergence. mixed conditional value-at-risk
JIMO
Stability analysis for set-valued vector mixed variational inequalities in real reflexive Banach spaces
Xing Wang Nan-Jing Huang
Journal of Industrial & Management Optimization 2013, 9(1): 57-74 doi: 10.3934/jimo.2013.9.57
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.
keywords: stability vector optimization problem upper semicontinuity boundedness nonemptiness Vector mixed variational inequality lower semicontinuity.
NACO
Strict feasibility for generalized mixed variational inequality in reflexive Banach spaces
Ren-You Zhong Nan-Jing Huang
Numerical Algebra, Control & Optimization 2011, 1(2): 261-274 doi: 10.3934/naco.2011.1.261
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 [28], 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 [8, 23].
keywords: generalized $f$-projection operator topological degree Generalized mixed variational inequality set-valued mapping.
JIMO
Some characterizations of the approximate solutions to generalized vector equilibrium problems
Yu Han Nan-Jing Huang
Journal of Industrial & Management Optimization 2016, 12(3): 1135-1151 doi: 10.3934/jimo.2016.12.1135
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.
keywords: upper semicontinuity. scalarization Generalized vector equilibrium problem approximate solution lower semicontinuity connectedness
JIMO
Neutral and indifference pricing with stochastic correlation and volatility
Jia Yue Nan-Jing Huang
Journal of Industrial & Management Optimization 2018, 14(1): 199-229 doi: 10.3934/jimo.2017043

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.

keywords: Wishart affine stochastic correlation utility-based pricing neutral and indifference pricing stochastic correlation and volatility
JIMO
Well-Posedness for vector quasi-equilibrium problems with applications
Nan-Jing Huang Xian-Jun Long Chang-Wen Zhao
Journal of Industrial & Management Optimization 2009, 5(2): 341-349 doi: 10.3934/jimo.2009.5.341
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.
keywords: vector quasi-optimization problem Vector quasi-equilibrium problem vector quasi-variational inequality problem approximating sequence. well-posedness
JIMO
Optimality conditions, duality and saddle points for nondifferentiable multiobjective fractional programs
Xian-Jun Long Nan-Jing Huang Zhi-Bin Liu
Journal of Industrial & Management Optimization 2008, 4(2): 287-298 doi: 10.3934/jimo.2008.4.287
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 literature.
keywords: Nondifferentiable multiobjective fractional programming d)$-convex function. saddle point \rho \alpha $(F Kuhn-Tucker optimality condition weakly efficient solution duality
JIMO
Levitin-Polyak well-posedness for variational inequalities and for optimization problems with variational inequality constraints
Rong Hu Ya-Ping Fang Nan-Jing Huang
Journal of Industrial & Management Optimization 2010, 6(3): 465-481 doi: 10.3934/jimo.2010.6.465
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.
keywords: metric characterization uniqueness. Levitin-Polyak well-posedness optimization problem with variational inequality constraints Variational inequality
NACO
Preface
Shengji Li Nan-Jing Huang Xinmin Yang
Numerical Algebra, Control & Optimization 2011, 1(3): i-ii doi: 10.3934/naco.2011.1.3i
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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