The paper shows that the global resolution of a general convex quadratic program with complementarity constraints (QPCC), possibly infeasible or unbounded, can be accomplished in finite time. The method constructs a minmax mixed integer formulation by introducing finitely many binary variables,...

This paper presents an algorithm and its implementation in the software package <ExternalRef> <RefSource> <Emphasis FontCategory="NonProportional">NCSOStools </RefSource> <RefTarget Address="http://ncsostools.fis.unm.si/" TargetType="URL"/> </ExternalRef> for finding sums of Hermitian squares and commutators decompositions for polynomials in noncommuting variables. The algorithm is based on noncommutative analogs of the classical Gram matrix method and...</emphasis></refsource></externalref>

We focus in this paper the problem of improving the semidefinite programming (SDP) relaxations for the standard quadratic optimization problem (standard QP in short) that concerns with minimizing a quadratic form over a simplex. We first analyze the duality gap between the standard QP and one of...

In this paper, we study iteration complexities of Mizuno-Todd-Ye predictor-corrector (MTY-PC) algorithms in SDP and symmetric cone programs by way of curvature integrals. The curvature integral is defined along the central path, reflecting the geometric structure of the central path. Integrating...

This paper addresses the general continuous single facility location problems in finite dimension spaces under possibly different <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$\ell _\tau $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi>ℓ</mi> <mi mathvariant="italic">τ</mi> </msub> </math> </EquationSource> </InlineEquation> norms, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$\tau \ge 1$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi mathvariant="italic">τ</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math> </EquationSource> </InlineEquation>, in the demand points. We analyze the difficulty of this family of problems and revisit...</equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation>

In a real situation, optimization problems often involve uncertain parameters. Robust optimization is one of distribution-free methodologies based on worst-case analyses for handling such problems. In this paper, we first focus on a special class of uncertain linear programs (LPs). Applying the...

We address the multi-period portfolio optimization problem with the constant rebalancing strategy. This problem is formulated as a polynomial optimization problem (POP) by using a mean-variance criterion. In order to solve the POPs of high degree, we develop a cutting-plane algorithm based on...

When using interior point methods for solving semidefinite programs (SDP), one needs to solve a system of linear equations at each iteration. For problems of large size, solving the system of linear equations can be very expensive. In this paper, we propose a trust region algorithm for solving...

We observe that in a simple one-dimensional polynomial optimization problem (POP), the ‘optimal’ values of semidefinite programming (SDP) relaxation problems reported by the standard SDP solvers converge to the optimal value of the POP, while the true optimal values of SDP relaxation...