Masakazu Kojima This paper is based on a recent work by Kojima which extended sums of squares relaxations of polynomial optimization problems to polynomial semidefinite programs. Let ${\cal E}$ and ${\cal E}_+$ be a finite dimensional real vector space and a symmetric cone embedded in ${\cal E}$; examples of $\calE$ and $\calE_+$ include a pair of the … Read more
A parallel computational method SDPARA-C is presented for SDPs (semidefinite programs). It combines two methods SDPARA and SDPA-C proposed by the authors who developed a software package SDPA. SDPARA is a parallel implementation of SDPA and it features parallel computation of the elements of the Schur complement equation system and a parallel Cholesky factorization of … Read more
A polynomial SDP (semidefinite program) minimizes a polynomial objective function over a feasible region described by a positive semidefinite constraint of a symmetric matrix whose components are multivariate polynomials. Sums of squares relaxations developed for polynomial optimization problems are extended to propose sums of squares relaxations for polynomial SDPs with an additional constraint for the … Read more
Representation of a given nonnegative multivariate polynomial in terms of a sum of squares of polynomials has become an essential subject in recent developments of sums of squares optimization and SDP (semidefinite programming) relaxation of polynomial optimization problems. We disscuss effective methods to obtain a simpler representation of a “sparse” polynomial as a sum of … Read more
Sequences of generalized Lagrangian duals and their SOS (sums of squares of polynomials) relaxations for a POP (polynomial optimization problem) are introduced. Sparsity of polynomials in the POP is used to reduce the sizes of the Lagrangian duals and their SOS relaxations. It is proved that the optimal values of the Lagrangian duals in the … Read more
The reliability of polyhedral homotopy continuation methods for solving a polynomial system becomes increasingly important as the dimension of the polynomial system increases. High powers of the homotopy continuation parameter $t$ and ill-conditioned Jacobian matrices encountered in tracing of homotopy paths affect the numerical stability. We present modified homotopy functions with a new homotopy continuation … Read more
PHoM is a software package in C++ for finding all isolated solutions of polynomial systems using a polyhedral homotopy continuation method. Among three modules constituting the package, the first module StartSystem constructs a family of polyhedral-linear homotopy functions, based on the polyhedral homotopy theory, from input data for a given system of polynomial equations $\f(\x) … Read more
Abstract: The SDPA (SemiDefinite Programming Algorithm) is known as efficient computer software based on primal-dual interior-point method for solving SDPs (Semidefinite Programs). In many applications, however, some SDPs become larger and larger, too large for the SDPA to solve on a single processor. In execution of the SDPA applied to large scale SDPs, the computation … Read more
The SDPA (SemiDefinite Programming Algorithm) is a software package for solving general SDPs(SemiDefinite Programs). It is written in C++ with the help of {\it LAPACK} for numerical linear algebra for dense matrix computation. The purpose of this paper is to present a brief description of the latest version of the SDPA and its high performance … Read more
The class of POPs (Polynomial Optimization Problems) over cones covers a wide range of optimization problems such as $0$-$1$ integer linear and quadratic programs, nonconvex quadratic programs and bilinear matrix inequalities. This paper presents a new framework for convex relaxation of POPs over cones in terms of linear optimization problems over cones. It provides a … Read more