A parametric active set method for quadratic programs with vanishing constraints

Combinatorial and logic constraints arising in a number of challenging optimization applications can be formulated as vanishing constraints. Quadratic programs with vanishing constraints (QPVCs) then arise as subproblems during the numerical solution of such problems using algorithms of the Sequential Quadratic Programming type. QPVCs are nonconvex problems violating standard constraint qualifications. In this paper, we … Read more

Methods for Convex and General Quadratic Programming

Computational methods are considered for finding a point that satisfies the second-order necessary conditions for a general (possibly nonconvex) quadratic program (QP). The first part of the paper considers the formulation and analysis of an active-set method for a generic QP with both equality and inequality constraints. The method uses a search direction that is … Read more

A Factorization with Update Procedures for a KKT Matrix Arising in Direct Optimal Control

Quadratic programs obtained for optimal control problems of dynamic or discrete–time processes usually involve highly block structured Hessian and constraints matrices. Efficient numerical methods for the solution of such QPs have to respect and exploit this block structure. In interior point methods, this is elegantly achieved by the widespread availability of advanced sparse symmetric indefinite … Read more

Continuous GRASP with a local active-set method for bound-constrained global optimization

Global optimization seeks a minimum or maximum of a multimodal function over a discrete or continuous domain. In this paper, we propose a hybrid heuristic – based on the CGRASP and GENCAN methods – for finding approximate solutions for continuous global optimization problems subject to box constraints. Experimental results illustrate the relative effectiveness of CGRASP-GENCAN … Read more

A Filter Active-Set Trust-Region Method

We develop a new active-set method for nonlinear programming problems that solves a regularized linear program to predict the active set and then fixes the active constraints to solve an equality-constrained quadratic program for fast convergence. Global convergence is promoted through the use of a filter. We show that the regularization parameter fulfills the same … Read more

An Active-Set Algorithm for Nonlinear Programming Using Parametric Linear Programming

This paper describes an active-set algorithm for nonlinear programming that solves a parametric linear programming subproblem at each iteration to generate an estimate of the active set. A step is then computed by solving an equality constrained quadratic program based on this active-set estimate. This approach respresents an extension of the standard sequential linear-quadratic programming … Read more

Global and finite termination of a two-phase augmented Lagrangian filter method for general quadratic programs

We present a two-phase algorithm for solving large-scale quadratic programs (QPs). In the first phase, gradient-projection iterations approximately minimize an augmented Lagrangian function and provide an estimate of the optimal active set. In the second phase, an equality-constrained QP defined by the current inactive variables is approximately minimized in order to generate a second-order search … Read more

Numerical methods for large-scale non-convex quadratic programming

We consider numerical methods for finding (weak) second-order critical points for large-scale non-convex quadratic programming problems. We describe two new methods. The first is of the active-set variety. Although convergent from any starting point, it is intended primarily for the case where a good estimate of the optimal active set can be predicted. The second … Read more