This is a progress report on an implementation of the active-set method for nonlinear programming proposed in [6] that employs piecewise linear models in the active-set prediction phase. The motivation for this work is to develop an algorithm that is capable of solving large-scale problems, including those with a large reduced space. Unlike SQP methods, … Read more
This paper describes an implementation of an interior-point algorithm for large-scale nonlinear optimization. It is based on the algorithm proposed by Curtis et al. (SIAM J Sci Comput 32:3447–3475, 2010), a method that possesses global convergence guarantees to first-order stationary points with the novel feature that inexact search direction calculations are allowed in order to … Read more
The complexity of finding epsilon-approximate first-order critical points for the general smooth constrained optimization problem is shown to be no worse that O(epsilon^{-2}) in terms of function and constraints evaluations. This result is obtained by analyzing the worst-case behaviour of a first-order shorts-step homotopy algorithm consisting of a feasibility phase followed by an optimization phase, … Read more
Accurate measurements of snow water equivalent (SWE) is an important factor in managing water resources for hydroelectric power generation. SWE over a catchment area may be estimated via kriging on measures obtained by snow monitoring devices positioned at strategic locations. The question studied in this paper is to find the device locations that minimize the … Read more
The classical trust-region subproblem (TRS) minimizes a nonconvex quadratic objective over the unit ball. In this paper, we consider extensions of TRS having extra constraints. When two parallel cuts are added to TRS, we show that the resulting nonconvex problem has an exact representation as a semidefinite program with additional linear and second-order-cone constraints. For … Read more
We consider a derivative-free optimization, and in particular black box optimization, where the functions to be minimized and the functions representing the constraints are given by black boxes without derivatives. Two fundamental families of methods are available: model-based methods and directional direct search algorithms. This work exploits the flexibility of the second type of methods … Read more
We describe a procedure to reduce variable bounds in Mixed Integer Nonlinear Programming (MINLP) as well as Mixed Integer Linear Programming (MILP) problems. The procedure works by combining pairs of inequalities of a linear programming (LP) relaxation of the problem. This bound reduction technique extends the implied bounds procedure used in MINLP and MILP and … Read more
Matrix rank minimization problems are gaining a plenty of recent attention in both mathematical and engineering fields. This class of problems, arising in various and across-discipline applications, is known to be NP-hard in general. In this paper, we aim at providing an approximation theory for the rank minimization problem, and prove that a rank minimization … Read more
The behavior of enumeration-based programs for solving MINLPs depends at least in part on the quality of the bounds on the optimal value and of the feasible solutions found. We consider MINLP problems with linear constraints. The convex hull relaxation (CHR) is a special case of the primal relaxation (Guignard 1994, 2007) that is very … Read more
We present an algorithm for nonlinear least-squares and nonlinear feasibility problems, i.e. for systems of nonlinear equations and nonlinear inequalities, which depend on the outcome of expensive functions for which derivatives are assumed to be unavailable. Our algorithm combines derivative-free techniques with filter trust-region methods to keep the number of expensive function evaluations low and … Read more