For the unconstrained optimization of black box functions, this paper introduces a new randomized algorithm called VRBBO. In practice, VRBBO matches the quality of other state-of-the-art algorithms for finding, in small and large dimensions, a local minimizer with reasonable accuracy. Although our theory guarantees only local minimizers our heuristic techniques turn VRBBO into an efficient … Read more
The convergence analysis of the alternating direction method of multipliers (ADMM) methods to convex/nonconvex combinational optimization have been well established in literature. Consider the extensive applications of weakly convex function in signal processing and machine learning(e.g. \textit{Special issue: DC-Theory, Algorithms and Applications, Mathematical Programming, 169(1):1-336,2018}), in this paper, we investigate the convergence analysis of ADMM … Read more
The main outcomes of the paper are divided into two parts. First, we present a new dual for quadratic programs, in which, the dual variables are affine functions, and we prove strong duality. Since the new dual is intractable, we consider a modified version by restricting the feasible set. This leads to a new bound … Read more
We construct norming meshes for polynomial optimization by the classical Markov inequality on general convex bodies in R^d, and by a tangential Markov inequality via an estimate of Dubiner distance on smooth convex bodies. These allow to compute a (1−eps)-approximation to the minimum of any polynomial of degree not exceeding n by O((n/sqrt(eps))^(ad)) samples, with … Read more
We construct web-shaped norming meshes on starlike polygons, by radial and boundary Chebyshev points. Via the approximation theoretic notion of Dubiner distance, we get a (1-eps)-approximation to the minimum of an arbitrary polynomial of degree n by O(n^2/eps) sampling points. CitationPreprint, July 2018 ArticleDownload View PDF
Two-trust-region subproblem (TTRS), which is the minimization of a general quadratic function over the intersection of two full-dimensional ellipsoids, has been the subject of several recent research. In this paper, to solve TTRS, a hybrid of efficient algorithms for finding global and local-nonglobal minimizers of trust-region subproblem and the alternating direction method of multipliers (ADMM) … Read more
In this paper, we consider the nonconvex quadratically constrained quadratic programming (QCQP) with one quadratic constraint. By employing the conjugate gradient method, an efficient algorithm is proposed to solve QCQP that exploits the sparsity of the involved matrices and solves the problem via solving a sequence of positive definite system of linear equations after identifying … Read more
Over the years, several classes of scalarization techniques in optimization have been introduced and employed in deriving separation theorems, optimality conditions and algorithms. In this paper, we study the relationships between some of those classes in the sense of inclusion. We focus on three types of scalarizing functionals (by Hiriart-Urruty, Drummond and Svaiter, Gerstewitz) and … Read more
We consider global optimization of nonconvex problems whose factorable reformulations contain a collection of multilinear equations. Important special cases include multilinear and polynomial optimization problems. The multilinear polytope is the convex hull of a set of binary points satisfying a number of multilinear equations. Running intersection inequalities are a family of facet-defining inequalities for the … Read more
It has recently been shown that the problem of testing global convexity of polynomials of degree four is {strongly} NP-hard, answering an open question of N.Z. Shor. This result is minimal in the degree of the polynomial when global convexity is of concern. In a number of applications however, one is interested in testing convexity … Read more