Approximation of the weighted maximin dispersion problem over Lp-ball: SDP relaxation is misleading

Consider the problem of finding a point in a unit $n$-dimensional $\ell_p$-ball ($p\ge 2$) such that the minimum of the weighted Euclidean distance from given $m$ points is maximized. We show in this paper that the recent SDP-relaxation-based approximation algorithm [SIAM J. Optim. 23(4), 2264-2294, 2013] will not only provide the first theoretical approximation bound of $\frac{1-O\left(\sqrt{ \ln(m)/n}\right)}{2}$, but also perform much better in practice, if the SDP relaxation is removed and the optimal solution of the SDP relaxation is replaced by a simple scalar matrix.Comment: 8pages,2figure

A new semidefinite relaxation for ℓ1\ell_{1}-constrained quadratic optimization and extensions

In this paper, by improving the variable-splitting approach, we propose a new semidefinite programming (SDP) relaxation for the nonconvex quadratic optimization problem over the $\ell_1$ unit ball (QPL1). It dominates the state-of-the-art SDP-based bound for (QPL1). As extensions, we apply the new approach to the relaxation problem of the sparse principal component analysis and the nonconvex quadratic optimization problem over the $\ell_p$ ($1< p<2$) unit ball and then show the dominance of the new relaxation.Comment: 13pages,1figur

Global Solutions to Large-Scale Spherical Constrained Quadratic Minimization via Canonical Dual Approach

This paper presents global optimal solutions to a nonconvex quadratic minimization problem over a sphere constraint. The problem is well-known as a trust region subproblem and has been studied extensively for decades. The main challenge is the so called 'hard case', i.e., the problem has multiple solutions on the boundary of the sphere. By canonical duality theory, this challenging problem is able to reformed as an one-dimensional canonical dual problem without duality gap. Sufficient and necessary conditions are obtained by the triality theory, which can be used to identify whether the problem is hard case or not. A perturbation method and the associated algorithms are proposed to solve this hard case problem. Theoretical results and methods are verified by large-size examples

Least Square Approximations and Linear Values of Cooperative Games

Many important values for cooperative games are known to arise from least square optimization problems. The present investigation develops an optimization framework to explain and clarify this phenomenon in a general setting. The main result shows that every linear value results from some least square approximation problem and that, conversely, every least square approximation problem with linear constraints yields a linear value. This approach includes and extends previous results on so-called least square values and semivalues in the literature. In particular, is it demonstrated how known explicit formulas for solutions under additional assumptions easily follow from the general results presented here

The Problem of Projecting the Origin of Euclidean Space onto the Convex Polyhedron

This paper is aimed at presenting a systematic survey of the existing now different formulations for the problem of projection of the origin of the Euclidean space onto the convex polyhedron (PPOCP). In the present paper, there are investigated the reduction of the projection program to the problems of quadratic programming, maximin, linear complementarity, and nonnegative least squares. Such reduction justifies the opportunity of utilizing a much more broad spectrum of powerful tools of mathematical programming for solving PPOCP. The paper's goal is to draw the attention of a wide range of research at the different formulations of the projection problem, which remain largely unknown due to the fact that the papers (addressing the subject of concern) are published even though on the adjacent, but other topics, or only in the conference proceedings.Comment: 19 page

Relaxing nonconvex quadratic functions by multiple adaptive diagonal perturbations

The current bottleneck of globally solving mixed-integer (non-convex) quadratically constrained problem (MIQCP) is still to construct strong but computationally cheap convex relaxations, especially when dense quadratic functions are present. We propose a cutting surface procedure based on multiple diagonal perturbations to derive strong convex quadratic relaxations for nonconvex quadratic problem with separable constraints. Our resulting relaxation does not use significantly more variables than the original problem, in contrast to many other relaxations based on lifting. The corresponding separation problem is a highly structured semidefinite program (SDP) with convex but non-smooth objective. We propose to solve this separation problem with a specialized primal-barrier coordinate minimization algorithm. Computational results show that our approach is very promising. First, our separation algorithm is at least an order of magnitude faster than interior point methods for SDPs on problems up to a few hundred variables. Secondly, on nonconvex quadratic integer problems, our cutting surface procedure provides lower bounds of almost the same strength with the diagonal SDP bounds used by (Buchheim and Wiegele, 2013) in their branch-and-bound code Q-MIST, while our procedure is at least an order of magnitude faster on problems with dimension greater than 70. Finally, combined with (linear) projected RLT cutting planes proposed by (Saxena, Bonami and Lee, 2011), our procedure provides slightly weaker bounds than their projected SDP+RLT cutting surface procedure, but in several order of magnitude shorter time. Finally we discuss various avenues to extend our work to design more efficient branch-and-bound algorithms for MIQCPs.Comment: 23 pages, including Appendi

Some Stability Properties of Parametric Quadratically Constrained Nonconvex Quadratic Programs in Hilbert Spaces

Stability of nonconvex quadratic programming problems under finitely many convex quadratic constraints in Hilbert spaces is investigated. We present several stability properties of the global solution map, and the continuity of the optimal value function, assuming that the problem data undergoes small perturbations.Comment: accepted for publication in AM

Estimation of Hurst Parameter of Fractional Brownian Motion Using CMARS Method

In this study, we develop a new theory of estimating Hurst parame- ter using conic multivariate adaptive regression splines (CMARS) method. We concentrate on the strong solution of stochastic differentional equations (SDEs) driven by fractional Brownian motion (fBm). The superiority of our approach to the others is, it not only estimates the Hurst parameter but also finds spline parameters of the stochastic process in an adaptive way. We examine the performance of our estimations using simulated test data. Keywords: Stochastic differential equations, fractional Brownian motion, Hurst parameter, conic multivariate adaptive regression spline

A Newton-bracketing method for a simple conic optimization problem

For the Lagrangian-DNN relaxation of quadratic optimization problems (QOPs), we propose a Newton-bracketing method to improve the performance of the bisection-projection method implemented in BBCPOP [to appear in ACM Tran. Softw., 2019]. The relaxation problem is converted into the problem of finding the largest zero $y^*$ of a continuously differentiable (except at $y^*$) convex function $g : \mathbb{R} \rightarrow \mathbb{R}$ such that $g(y) = 0$ if $y \leq y^*$ and $g(y) > 0$ otherwise. In theory, the method generates lower and upper bounds of $y^*$ both converging to $y^*$. Their convergence is quadratic if the right derivative of $g$ at $y^*$ is positive. Accurate computation of $g'(y)$ is necessary for the robustness of the method, but it is difficult to achieve in practice. As an alternative, we present a secant-bracketing method. We demonstrate that the method improves the quality of the lower bounds obtained by BBCPOP and SDPNAL+ for binary QOP instances from BIQMAC. Moreover, new lower bounds for the unknown optimal values of large scale QAP instances from QAPLIB are reported.Comment: 19 pages, 2 figure

Isotonic regression and isotonic projection

The note describes the cones in the Euclidean space admitting isotonic metric projection with respect to the coordinate-wise ordering. As a consequence it is showed that the metric projection onto the regression cone (the cone defined by the general isotonic regression problem) admits a projection which is isotonic with respect to the coordinate-wise ordering.Comment: 10 page