A note on polynomial solvability of the CDT problem
We describe a simple polynomial-time algorithm for the CDT problems that relies on a construction of Barvinok. CitationColumbia UniversityArticleDownload View PDF
trust-region methods
A Two-Variable Approach to the Two-Trust-Region Subproblem
The trust-region subproblem minimizes a general quadratic function over an ellipsoid and can be solved in polynomial time using a semidefinite-programming (SDP) relaxation. Intersecting the feasible set with a second ellipsoid results in the two-trust-region subproblem (TTRS). Even though TTRS can also be solved in polynomial-time, existing algorithms do not use SDP. In this paper, … Read more
Narrowing the difficulty gap for the Celis-Dennis-Tapia problem
We study the {\em Celis-Dennis-Tapia (CDT) problem}: minimize a non-convex quadratic function over the intersection of two ellipsoids. In contrast to the well-studied trust region problem where the feasible set is just one ellipsoid, the CDT problem is not yet fully understood. Our main objective in this paper is to narrow the difficulty gap that … Read more
Polynomial solvability of variants of the trust-region subproblem
The trust region subproblem concerns the minimization of a general quadratic over the unit ball in R^n. Extensions to this problem are of interest because of applications to, for example, combinatorial optimization. However the extension obtained by adding an arbitrary family of linear side constraints is NP-hard. In this paper we consider variants of the … Read more
A Trust-Region Method for Unconstrained Multiobjective Problems with Applications in Satisficing Processes
Multiobjective optimization has a significant number of real life applications. For this reason, in this paper, we consider the problem of finding Pareto critical points for unconstrained multiobjective problems and present a trust-region method to solve it. Under certain assumptions, which are derived in a very natural way from assumptions used by \citet{conn} to establish … Read more
Convergence of trust-region methods based on probabilistic models
In this paper we consider the use of probabilistic or random models within a classical trust-region framework for optimization of deterministic smooth general nonlinear functions. Our method and setting differs from many stochastic optimization approaches in two principal ways. Firstly, we assume that the value of the function itself can be computed without noise, in … Read more
Globally convergent DC trust-region methods
In this paper, we investigate the use of DC (Difference of Convex functions) models and algorithms in the solution of nonlinear optimization problems by trust-region methods. We consider DC local models for the quadratic model of the objective function used to compute the trust-region step, and apply a primal-dual subgradient method to the solution of … Read more
The Trust Region Subproblem with Non-Intersecting Linear Constraints
This paper studies an extended trust region subproblem (eTRS)in which the trust region intersects the unit ball with m linear inequality constraints. When m=0, m=1, or m=2 and the linear constraints are parallel, it is known that the eTRS optimal value equals the optimal value of a particular convex relaxation, which is solvable in polynomial … Read more
MSS: MATLAB software for L-BFGS trust-region subproblems for large-scale optimization
A MATLAB implementation of the More’-Sorensen sequential (MSS) method is presented. The MSS method computes the minimizer of a quadratic function defined by a limited-memory BFGS matrix subject to a two-norm trust-region constraint. This solver is an adaptation of the More’-Sorensen direct method into an L-BFGS setting for large-scale optimization. The MSS method makes use … Read more
Adaptive Regularized Self-Consistent Field Iteration with Exact Hessian for Electronic Structure Calculation
The self-consistent field (SCF) iteration has been used ubiquitously for solving the Kohn-Sham (KS) equation or the minimization of the KS total energy functional with respect to orthogonality constraints in electronic structure calculations. Although SCF with heuristics such as charge mixing often works remarkably well on many problems, it is well known that its convergence … Read more