Nonlinear Optimization

Iterative Reweighted Linear Least Squares for Exact Penalty Subproblems on Product Sets

Categories Constrained Nonlinear Optimization, Convex Optimization, Nonlinear Optimization Tags , , , , ,

We present two matrix-free methods for solving exact penalty subproblems on product sets that arise when solving large-scale optimization problems. The first approach is a novel iterative reweighting algorithm (IRWA), which iteratively minimizes quadratic models of relaxed subproblems while automatically updating a relaxation vector. The second approach is based on alternating direction augmented Lagrangian (ADAL) … Read more

An Interior-Point Trust-Funnel Algorithm for Nonlinear Optimization

Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , , , , ,

We present an interior-point trust-funnel algorithm for solving large-scale nonlinear optimization problems. The method is based on an approach proposed by Gould and Toint (Math Prog 122(1):155–196, 2010) that focused on solving equality constrained problems. Our method is similar in that it achieves global convergence guarantees by combining a trust-region methodology with a funnel mechanism, … Read more

Preconditioning issues in the numerical solution of nonlinear equations and nonlinear least squares

Categories Nonlinear Optimization Tags , ,

Second order methods for optimization call for the solution of sequences of linear systems. In this survey we will discuss several issues related to the preconditioning of such sequences. Covered topics include both techniques for building updates of factorized preconditioners and quasi-Newton approaches. Sequences of unsymmetric linear systems arising in Newton- Krylov methods will be … Read more

The inexact projected gradient method for quasiconvex vector optimization problems

Categories Generalized Convexity/Monoticity, Multi-Criteria Optimization, Nonlinear Optimization

Vector optimization problems are a generalization of multiobjective optimization in which the preference order is related to an arbitrary closed and convex cone, rather than the nonnegative octant. Due to its real life applications, it is important to have practical solution approaches for computing. In this work, we consider the inexact projected gradient-like method for … Read more

A Relaxed-Projection Splitting Algorithm for Variational Inequalities in Hilbert Spaces

Categories Complementarity and Variational Inequalities, Nonlinear Optimization, Nonsmooth Optimization Tags , , , , ,

We introduce a relaxed-projection splitting algorithm for solving variational inequalities in Hilbert spaces for the sum of nonsmooth maximal monotone operators, where the feasible set is defined by a nonlinear and nonsmooth continuous convex function inequality. In our scheme, the orthogonal projections onto the feasible set are replaced by projections onto separating hyperplanes. Furthermore, each … Read more

Lagrangean Decomposition for Mean-Variance Combinatorial Optimization

Categories Combinatorial Optimization, Constrained Nonlinear Optimization, Robust Optimization Tags , ,

We address robust versions of combinatorial optimization problems, focusing on the uncorrelated ellipsoidal uncertainty case, which corresponds to so-called mean-variance optimization. We present a branch and bound-algorithm for such problems that uses lower bounds obtained from Lagrangean decomposition. This approach allows to separate the uncertainty aspect in the objective function from the combinatorial structure of … Read more

Active Set Methods with Reoptimization for Convex Quadratic Integer Programming

Categories (Mixed) Integer Nonlinear Programming, Convex Optimization, Quadratic Programming Tags , ,

We present a fast branch-and-bound algorithm for solving convex quadratic integer programs with few linear constraints. In each node, we solve the dual problem of the continuous relaxation using an infeasible active set method proposed by Kunisch and Rendl to get a lower bound; this active set algorithm is well suited for reoptimization. Our algorithm … Read more

Penalty Methods with Stochastic Approximation for Stochastic Nonlinear Programming

Categories Nonlinear Optimization, Stochastic Programming

In this paper, we propose a class of penalty methods with stochastic approximation for solving stochastic nonlinear programming problems. We assume that only noisy gradients or function values of the objective function are available via calls to a stochastic first-order or zeroth-order oracle. In each iteration of the proposed methods, we minimize an exact penalty … Read more

A Revisit to Quadratic Programming with One Inequality Quadratic Constraint via Matrix Pencil

Categories Global Optimization Theory, Quadratic Programming Tags , , , , ,

The quadratic programming over one inequality quadratic constraint (QP1QC) is a very special case of quadratically constrained quadratic programming (QCQP) and attracted much attention since early 1990’s. It is now understood that, under the primal Slater condition, (QP1QC) has a tight SDP relaxation (PSDP). The optimal solution to (QP1QC), if exists, can be obtained by … Read more