Let Box_n = {x in R^n : 0<= x <= e }, and let QPB_n denote the convex hull of {(1, x’)'(1, x’) : x in Box_n}. The quadratic programming problem min{x’Q x + q’x : x in Box_n} where Q is not positive semidefinite (PSD), is equivalent to a linear optimization problem over QPB_n … Read more
The problem considered is a multi-objective optimization problem, in which the goal is to find an optimal value of a vector function representing various criteria. The aim of this work is to develop an algorithm which utilizes the trust region framework with probabilistic model functions, able to cope with noisy problems, using inaccurate functions and … Read more
Constraint qualifications (CQs) are fundamental for understanding the geometry of feasible sets and for ensuring the validity of optimality conditions in nonlinear programming. A known idea is that constant-rank type CQs allow one to modify the description feasible set, by eliminating redundant constraints, so that the Mangasarian-Fromovitz CQ (MFCQ) holds. Traditionally, such modifications, called reductions … Read more
We consider trust-region methods for solving optimization problems where the objective is the sum of a smooth, nonconvex function and a nonsmooth, convex regularizer. We extend the global convergence theory of such methods to include worst-case complexity bounds in the case of unbounded model Hessian growth, and introduce a new, simple nonsmooth trust-region subproblem solver … Read more
This paper deals with the Constrained Riemannian Optimization (CRO) problem, which involves minimizing a function subject to equality and inequality constraints on Riemannian manifolds. The study aims to advance optimization theory in the Riemannian setting by presenting and analyzing a penalty-type method for solving CRO problems. The proposed approach is based on techniques that involve … Read more
This paper presents a novel restarted version of Nesterov’s accelerated gradient method and establishes its optimal iteration-complexity for solving convex smooth composite optimization problems. The proposed restart accelerated gradient method is shown to be a specific instance of the accelerated inexact proximal point framework introduced in “An accelerated hybrid proximal extragradient method for convex optimization … Read more
We consider the problem of computing the optimal solution and objective of a linear program under linearly changing linear constraints. The problem studied is given by $\min c^t x \text{ s.t } Ax + \lambda Dx \leq b$ where $\lambda$ belongs to a set of predefined values $\Lambda$. Based on the information given by a … Read more
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Inspired by the impact of the Goemans-Williamson algorithm on combinatorial optimization, we construct an analogous relax-then-round strategy for low-rank optimization problems. First, for orthogonally constrained quadratic optimization problems, we derive a semidefinite relaxation and a randomized rounding scheme that obtains provably near-optimal solutions, building on the blueprint from Goemans and Williamson for the Max-Cut problem. … Read more
We introduce two-stage stochastic min-max and min-min integer programs with bi-parameterized recourse (BTSPs), where the first-stage decisions affect both the objective function and the feasible region of the second-stage problem. To solve these programs efficiently, we introduce Lagrangian-integrated \(L\)-shaped (\(L^2\)) methods, which guarantee exact solutions when the first-stage decisions are pure binary. For mixed-binary first-stage … Read more