Xiaoqi Yang

A Unified Augmented Lagrangian Approach to Duality and Exact Penalization

In this paper, the existence of an optimal path and its convergence to the optimal set of a primal problem of minimizing an extended real-valued function are established via a generalized augmented Lagrangian and corresponding generalized augmented Lagrangian problems, in which no convexity is imposed on the augmenting function. These results further imply a zero duality gap property between the primal problem and the generalized augmented Lagrangian dual problem. A necessary and sufficient condition for the exact penalty representation in the framework of a generalized augmented Lagrangian is obtained. In the context of constrained programs, we show that generalized augmented Lagrangians present a unified approach to several classes of exact penalization results. Some equivalences among exact penalization results are obtained.

A Nonlinear Lagrangian Approach to Constrained Optimization Problems

In this paper we study nonlinear Lagrangian functions for constrained optimization problems which are, in general, nonlinear with respect to the objective function. We establish an equivalence between two types of zero duality gap properties, which are described using augmented Lagrangian dual functions and nonlinear Lagrangian dual functions, respectively. Furthermore, we show the existence of a path of optimal solutions generated by nonlinear Lagrangian problems and show its convergence toward the optimal set of the original problem. We analyze the convergence of several classes of nonlinear Lagrangian problems in terms of their first and second order necessary optimality conditions.

Second-order global optimality conditions for convex composite optimization

In recent years second-order sufficient conditions of an isolated local minimizer for convex composite optimization problems have been established. In this paper, second-order optimality conditions are obtained of aglobal minimizer for convex composite problems with a non-finite valued convex function and a twice strictly differentiable function by introducing a generalized representation condition. This result is applied to a minimization problem with a closed convex set constraint which is shown to satisfy the basic constraint qualification. In particular, second-order necessary and sufficient conditions of a solution for a variational inequality problem with convex composite inequality constraints are obtained.

Near-field broadband beamformer design via multidimensional semi-infinite-linear programming techniques

Broadband microphone arrays has important applications such as hands-free mobile telephony, voice interface to personal computers and video conference equipment. This problem can be tackled in different ways. In this paper, a general broadband beamformer design problem is considered. The problem is posed as a Chebyshev minimax problem. Using the l/sub 1/-norm measure or the real rotation theorem, we show that it can be converted into a semi-infinite linear programming problem. A numerical scheme using a set of adaptive grids is applied. The scheme is proven to be convergent when a certain grid refinement is used. The method can be applied to the design of multidimensional digital finite-impulse response (FIR) filters with arbitrarily specified amplitude and phase.

Extended Lagrange And Penalty Functions in Continuous Optimization

We consider problems of continuous constrainedoptimization in finite dimensional space and studygeneralized nonlinear Lagrangian and penaltyfunctions which are formed by increasing convolutionfunctions

Vector equilibrium problem and vector optimization

This paper examines the vector equilibrium model based on a vector cost consideration. This is a generalization of the well-known Wardrop traffic equilibrium principle where road users choose paths based on just a single cost. The concept of parametric equilibria is introduced and used to establish relations with parametric complementarity and variational inequality problems. Relations with some vector optimization problems via scalarization techniques are given under appropriate conditions. Some solution methods for solving vector equilibrium problems are also discussed.

Lagrange Multipliers in Nonsmooth Semi-Infinite Optimization Problems

Using the variational analysis technique, in terms of the epi-coderivative, we provide Lagrange multiplier rules for a class of semi-infinite optimization problems where all functions are lower semicontinuous or locally Lipschitz.

On the Smoothing of the Square-Root Exact Penalty Function for Inequality Constrained Optimization

In this paper we propose two methods for smoothing a nonsmooth square-root exact penalty function for inequality constrained optimization. Error estimations are obtained among the optimal objective function values of the smoothed penalty problem, of the nonsmooth penalty problem and of the original optimization problem. We develop an algorithm for solving the optimization problem based on the smoothed penalty function and prove the convergence of the algorithm. The efficiency of the smoothed penalty function is illustrated with some numerical examples, which show that the algorithm seems efficient.

Weak Sharp Minima for Semi-infinite Optimization Problems with Applications

We study local weak sharp minima and sharp minima for smooth semi-infinite optimization problems SIP. We provide several dual and primal characterizations for a point to be a sharp minimum or a weak sharp minimum of SIP. As applications, we present several sufficient and necessary conditions of calmness for infinitely many smooth inequalities. In particular, we improve some calmness results in [R. Henrion and J. Outrata, Math. Program., 104 (2005), pp. 437-464].

Weak Sharp Minima in Multicriteria Linear Programming

In this short note, we study the existence of weak sharp minima in multicriteria linear programming. It is shown that weak sharp minimality holds for certain residual functions and gap functions.