JIMO
A barrier function method for generalized Nash equilibrium problems
Jian Hou Liwei Zhang
Journal of Industrial & Management Optimization 2014, 10(4): 1091-1108 doi: 10.3934/jimo.2014.10.1091
In this paper, we propose a barrier function method for the generalized Nash equilibrium problem (GNEP) which, in contrast to the standard Nash equilibrium problem (NEP), allows the constraints for each player may depend on the rivals' strategies. We solve a sequence of NEPs, which are defined by logarithmic barrier functions of the joint inequality constraints. We demonstrate, under suitable conditions, that any accumulation point of the solutions to the sequence of NEPs is a solution to the GNEP. Moreover, a semismooth Newton method is used to solve the NEPs and sufficient conditions for the local superlinear convergence rate of the semismooth Newton method are derived. Finally, numerical results are reported to illustrate that the barrier approach for solving the GNEP is practical.
keywords: generalized Nash equilibrium problem logarithmic barrier function semismooth Newton method. Nash equilibrium problem quasi-variational inequalities
JIMO
A nonlinear Lagrangian method based on Log-Sigmoid function for nonconvex semidefinite programming
Yang Li Liwei Zhang
Journal of Industrial & Management Optimization 2009, 5(3): 651-669 doi: 10.3934/jimo.2009.5.651
We present a nonlinear Lagrangian method for nonconvex semidefinite programming. This nonlinear Lagrangian is generated by a Löwner operator associated with Log-Sigmoid function. Under a set of assumptions, we prove a convergence theorem, which shows that the nonlinear Lagrangian algorithm is locally convergent when the penalty parameter is less than a threshold and the error bound of the solution is proportional to the penalty parameter.
keywords: Löwner operator. nonlinear Lagrangian nonconvex semidefinite programming
JIMO
A sequential convex program method to DC program with joint chance constraints
Xiantao Xiao Jian Gu Liwei Zhang Shaowu Zhang
Journal of Industrial & Management Optimization 2012, 8(3): 733-747 doi: 10.3934/jimo.2012.8.733
In this paper, we consider a DC (difference of convex) programming problem with joint chance constraints (JCCDCP). We propose a DC function to approximate the constrained function and a corresponding DC program ($\textrm{P}_{\varepsilon}$) to approximate the JCCDCP. Under some mild assumptions, we show that the solution of Problem ($\textrm{P}_{\varepsilon}$) converges to the solution of JCCDCP when $\varepsilon\downarrow 0$. A sequential convex program method is constructed to solve the Problem ($\textrm{P}_{\varepsilon}$). At each iteration a convex program is solved based on the Monte Carlo method, and the generated optimal sequence is proved to converge to the stationary point of Problem ($\textrm{P}_{\varepsilon}$).
keywords: joint chance constraints. DC programming Sequential convex approximation approach
JIMO
Perturbation analysis of a class of conic programming problems under Jacobian uniqueness conditions
Ziran Yin Liwei Zhang
Journal of Industrial & Management Optimization 2018, 13(5): 1-11 doi: 10.3934/jimo.2018100

We consider the stability of a class of parameterized conic programming problems which are more general than $C^2$-smooth parameterization. We show that when the Karush-Kuhn-Tucker (KKT) condition, the constraint nondegeneracy condition, the strict complementary condition and the second order sufficient condition (named as Jacobian uniqueness conditions here) are satisfied at a feasible point of the original problem, the Jacobian uniqueness conditions of the perturbed problem also hold at some feasible point.

keywords: Stability $C^2$-cone reducible sets Jacobian uniqueness conditions KKT system implicit function theorem
JIMO
A perturbation approach for an inverse linear second-order cone programming
Yi Zhang Yong Jiang Liwei Zhang Jiangzhong Zhang
Journal of Industrial & Management Optimization 2013, 9(1): 171-189 doi: 10.3934/jimo.2013.9.171
A type of inverse linear second-order cone programming problems is discussed, in which the parameters in both the objective function and the constraint set of a given linear second-order cone programming need to be adjusted as little as possible so that a known feasible solution becomes optimal. This inverse problem can be formulated as a minimization problem with second-order cone complementarity constraints. With the help of the smoothed Fischer-Burmeister function over second-order cones, we construct a smoothing approximation of the formulated problem whose feasible set and optimal solution set are demonstrated to be continuous and outer semicontinuous respectively at the perturbed parameter $\varepsilon=0$. A damped Newton method is employed to solve the perturbed problem and its global convergence and local quadratic convergence rate are shown. Finally, the numerical results are reported to show the effectiveness of the damped Newton method for solving the inverse linear second-order cone programming.
keywords: damped Newton method. Inverse optimization linear second-order cone programming perturbation approach
JIMO
A smoothing augmented Lagrangian method for nonconvex, nonsmooth constrained programs and its applications to bilevel problems
Qingsong Duan Mengwei Xu Yue Lu Liwei Zhang
Journal of Industrial & Management Optimization 2018, 13(5): 1-21 doi: 10.3934/jimo.2018094

In this paper, we consider a class of nonsmooth and nonconvex optimization problem with an abstract constraint. We propose an augmented Lagrangian method for solving the problem and construct global convergence under a weakly nonsmooth Mangasarian-Fromovitz constraint qualification. We show that any accumulation point of the iteration sequence generated by the algorithm is a feasible point which satisfies the first order necessary optimality condition provided that the penalty parameters are bounded and the upper bound of the augmented Lagrangian functions along the approximated solution sequence exists. Numerical experiments show that the algorithm is efficient for obtaining stationary points of general nonsmooth and nonconvex optimization problems, including the bilevel program which will never satisfy the nonsmooth Mangasarian-Fromovitz constraint qualification.

keywords: Nonsmooth constrained problem smoothing function augmented Lagrangian method constraint qualification bilevel program
NACO
A class of smoothing SAA methods for a stochastic linear complementarity problem
Jie Zhang Yue Wu Liwei Zhang
Numerical Algebra, Control & Optimization 2012, 2(1): 145-156 doi: 10.3934/naco.2012.2.145
A class of smoothing sample average approximation (SAA) methods is proposed for solving a stochastic linear complementarity problem, where the underlying function is the expected value of stochastic function. Existence and convergence results to the proposed methods are provided and some numerical results are reported to show the efficiency of the methods proposed.
keywords: stochastic linear complementarity problem. Smoothing SAA method
JIMO
A differential equation method for solving box constrained variational inequality problems
Li Wang Yang Li Liwei Zhang
Journal of Industrial & Management Optimization 2011, 7(1): 183-198 doi: 10.3934/jimo.2011.7.183
In this paper, we discuss a system of differential equations based on the projection operator for solving the box constrained variational inequality problems. The equilibrium solutions to the differential equation system are proved to be the solutions of the box constrained variational inequality problems. Two differential inclusion problems associated with the system of differential equations are introduced. It is proved that the equilibrium solution to the differential equation system is locally asymptotically stable by verifying the locally asymptotical stability of the equilibrium positions of the differential inclusion problems. An Euler discrete scheme with Armijo line search rule is introduced and its global convergence is demonstrated. The numerical experiments are reported to show that the Euler method is effective.
keywords: global convergence. differential inclusion asymptotical stability differential equations Box constrained variational inequality problem
JIMO
A penalty method for generalized Nash equilibrium problems
Yanhong Yuan Hongwei Zhang Liwei Zhang
Journal of Industrial & Management Optimization 2012, 8(1): 51-65 doi: 10.3934/jimo.2012.8.51
This paper considers a penalty algorithm for solving the generalized Nash equilibrium problem (GNEP). Under the GNEP-MFCQ at a limiting point of the sequence generated by the algorithm, we demonstrate that the limit point is a solution to the GNEP and the parameter becomes a constant after a finite iterations. We formulate the Karush-Kuhn-Tucker conditions for a penalized problem into a system of nonsmooth equations and demonstrate the nonsingularity of its Clarke’s generalized Jacobian at the KKT point under the so-called GNEP-Second Order Sufficient Condition. The nonsingularity facilitates the application of the smoothing Newton method for the global convergence and local quadratic rate. Finally, the numerical results are reported to show the effectiveness of the penalty method in solving generalized Nash equilibrium problem.
keywords: Clarke’s generalized Jacobian. smoothing Newton method Generalized Nash equilibrium penalty method
JIMO
Second-order optimality conditions for cone constrained multi-objective optimization
Liwei Zhang Jihong Zhang Yule Zhang
Journal of Industrial & Management Optimization 2018, 14(3): 1041-1054 doi: 10.3934/jimo.2017089

The aim of this paper is to develop second-order necessary and second-order sufficient optimality conditions for cone constrained multi-objective optimization. First of all, we derive, for an abstract constrained multi-objective optimization problem, two basic necessary optimality theorems for weak efficient solutions and a second-order sufficient optimality theorem for efficient solutions. Secondly, basing on the optimality results for the abstract problem, we demonstrate, for cone constrained multi-objective optimization problems, the first-order and second-order necessary optimality conditions under Robinson constraint qualification as well as the second-order sufficient optimality conditions under upper second-order regularity for the conic constraint. Finally, using the optimality conditions for cone constrained multi-objective optimization obtained, we establish optimality conditions for polyhedral cone, second-order cone and semi-definite cone constrained multi-objective optimization problems.

keywords: Cone constrained multi-objective optimization second-order optimality conditions polyhedral cone second-order cone semi-definite cone

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