Some properties of regularized and penalized nonlinear programming formulations of mathematical programs with equilibrium constraints (MPECs) are described. The focus is on the properties of these formulations near a local solution of the MPEC at which strong stationarity and a second-order sufficient condition are satisfied. In the regularized formulations, the complementarity condition is replaced by … Read more
A mixed interior/exterior-point method for nonlinear programming is described, that handles constraints by an L1-penalty function. A suitable decomposition of the penalty terms and embedding of the problem into a higher-dimensional setting leads to an equivalent, surprisingly regular, reformulation as a smooth penalty problem only involving inequality constraints. The resulting problem may then be tackled … Read more
In this paper we consider the question of solving equilibrium problems—formulated as complementarity problems and, more generally, mathematical programs with equilibrium constraints (MPEC’s)—as nonlinear programs, using an interior-point approach. These problems pose theoretical difficulties for nonlinear solvers, including interior-point methods. We examine the use of penalty methods to get around these difficulties, present an example … Read more
The paper describes and analyzes an algorithmic framework for solving nonlinear programming problems in which strict complementarity conditions and constraint qualifications are not necessarily satisfied at a solution. The framework is constructed from three main algorithmic ingredients. The first is any conventional method for nonlinear programming that produces estimates of the Lagrange multipliers at each … Read more
A hybrid interior-point method for nonlinear programming is presented. It enjoys the flexibility of switching between a line search based method which computes steps by factoring the primal-dual equations and an iterative method using a conjugate gradient algorithm and globalized by means of trust regions. Steps computed by a direct factorization are always tried first, … Read more
Interior point methods for nonlinear programs (NLPs) are adapted for solution of mathematical programs with complementarity constraints (MPCCs). The constraints of the MPCC are suitably relaxed so as to guarantee a strictly feasible interior for the inequality constraints. The standard primal-dual algorithm has been adapted with a modified step calculation. The algorithm is shown to … Read more
This paper presents a strategy for choosing the initial point, slacks and multipliers in interior methods for nonlinear programming. It consists of first computing a Newton-like step to estimate the magnitude of these three variables and then shifting the slacks and multipliers so that they are sufficiently positive. The new strategy has the option of … Read more
We transform the system of nonlinear equations into a nonlinear programming problem, which is solved by null space algorithms. We do not use standard least square approach. We divide the equations into two groups. One group contains the equations that are treated as equality constraints. The square of other equations is regarded as objective function. … Read more
Filter approach is recently proposed by Fletcher and Leyffer in 2002 and is attached importance to. In this paper, the filter approach is used in an sequential penalty quadratic programming (S$l$QP) algorithm which is similar to that of Yuan’s. In every trial step, the step length is controlled by a trust region radius. If the … Read more
In this paper the theory of local convergence for a class of line search filter type methods for nonlinear programming is presented. The algorithm presented here is globally convergent (see Chin [4]) and the rate of convergence is two-step superlinear. The proposed algorithm solves a sequence of quadratic progrmming subproblems to obtain search directions and … Read more