# American Institute of Mathematical Sciences

## Journals

JIMO
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
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
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
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
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
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

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
NACO
We consider a type of generalized Nash equilibrium problems with second-order cone constraints. The Karush-Kuhn-Tucker system can be formulated as a system of semismooth equations involving metric projectors. Furthermore, the smoothing Newton method is given to get a Karush-Kuhn-Tucker point of the problem. The nonsingularity of Clarke's generalized Jacobian of the Karush-Kuhn-Tucker system, which is needed in the convergence analysis of smoothing Newton method, is demonstrated under the so-called constraint nondegeneracy condition in generalized Nash equilibrium problems and pseudo-strong second order optimality condition. At last, we take some experiments, in which the smoothing Newton method is applied. Furthermore, we get the normalized equilibria in the constraint-shared case. The numerical results show that the smoothing Newton method has a good performance in solving this type of generalized Nash equilibrium problems.
keywords: Generalized Nash equilibrium second-order cone smoothing Newton method. metric projector
JIMO

This paper focuses on a class of mathematical programs with symmetric cone complementarity constraints (SCMPCC). The explicit expression of C-stationary condition and SCMPCC-linear independence constraint qualification (denoted by SCMPCC-LICQ) for SCMPCC are first presented. We analyze a parametric smoothing approach for solving this program in which SCMPCC is replaced by a smoothing problem $P_{\varepsilon}$ depending on a (small) parameter $\varepsilon$. We are interested in the convergence behavior of the feasible set, stationary points, solution mapping and optimal value function of problem $P_{\varepsilon}$ when $\varepsilon \to 0$ under SCMPCC-LICQ. In particular, it is shown that the convergence rate of Hausdorff distance between feasible sets $\mathcal{F}_{\varepsilon}$ and $\mathcal{F}$ is of order $\mbox{O}(|\varepsilon|)$ and the solution mapping and optimal value of $P_{\varepsilon}$ are outer semicontinuous and locally Lipschitz continuous at $\varepsilon=0$ respectively. Moreover, any accumulation point of stationary points of $P_{\varepsilon}$ is a C-stationary point of SCMPCC under SCMPCC-LICQ.

keywords: Mathematical program with symmetric cone complementarity constraints C-stationary point parametric smoothing approach rate of convergence
JIMO
We consider an inverse problem raised from the semi-definite quadratic programming (SDQP) problem. In the inverse problem, the parameters in the objective function of a given SDQP problem are adjusted as little as possible so that a known feasible solution becomes the optimal one. We formulate this problem as a minimization problem with a positive semi-definite cone constraint and its dual is a linearly positive semi-definite cone constrained semismoothly differentiable ($\mbox{SC}^1$) convex programming problem with fewer variables than the original one. We demonstrate the global convergence of the augmented Lagrangian method for the dual problem and prove that the convergence rate of primal iterates, generated by the augmented Lagrange method, is proportional to $1/t$, and the rate of multiplier iterates is proportional to $1/\sqrt{t}$, where $t$ is the penalty parameter in the augmented Lagrangian. The numerical results are reported to show the effectiveness of the augmented Lagrangian method for solving the inverse semi-definite quadratic programming problem.
keywords: quadratic programming Inverse optimization Newton method. the augmented Lagrangian method rate of convergence the cone of positive semi-definite matrices