We present two algorithms for large-scale low-rank Euclidean distance matrix completion problems, based on semidefinite optimization. Our first method works by relating cliques in the graph of the known distances to faces of the positive semidefinite cone, yielding a combinatorial procedure that is provably robust and parallelizable. Our second algorithm is a first order method … Read more
Interior-point methods are known to be sensitive to ill-conditioning and to scaling of the data. This paper presents new asymptotically sharp bounds on the condition numbers of the linear systems at each iteration of an interior-point method for solving linear or semidefinite programs and discusses a stopping test which leads to a problem-independent “a priori” … Read more
Given a univariate polynomial, its abscissa is the maximum real part of its roots. The abscissa arises naturally when controlling linear differential equations. As a function of the polynomial coefficients, the abscissa is H\”older continuous, and not locally Lipschitz in general, which is a source of numerical difficulties for designing and optimizing control laws. In … Read more
This paper is concerned with completely positive and semidefinite relaxations of quadratic programs with linear constraints and binary variables as presented by Burer. It observes that all constraints of the relaxation associated with linear constraints of the original problem can be accumulated in a single linear constraint without changing the feasible set of either the … Read more
In this note, we prove that the KKT mapping for nonlinear semidefinite optimization problem is upper Lipschitz continuous at the KKT point, under the second-order sufficient optimality conditions and the strict Robinson constraint qualification. Article Download View On the upper Lipschitz property of the KKT mapping for nonlinear semidefinite optimization
We suppose the existence of an oracle which solves any semidefinite programming (SDP) problem satisfying Slater’s condition simultaneously at its primal and dual sides. We note that such an oracle might not be able to directly solve general SDPs even after certain regularization schemes are applied. In this work we fill this gap and show … Read more
We describe simple and exact duals, and certificates of infeasibility and weak infeasibility in conic linear programming which do not rely on any constraint qualification, and retain most of the simplicity of the Lagrange dual. In particular, some of our infeasibility certificates generalize the row echelon form of a linear system of equations, and the … Read more
We develop a new constraint-reduced infeasible predictor-corrector interior point method for semidefinite programming, and we prove that it has polynomial global convergence and superlinear local convergence. While the new algorithm uses HKM direction in predictor step, it adopts AHO direction in corrector step to obtain faster approach to the central path. In contrast to the … Read more
We consider the linear or quadratic 0/1 program \[P:\quad f^*=\min\{ c^Tx+x^TFx : \:A\,x =\b;\:x\in\{0,1\}^n\},\] for some vectors $c\in R^n$, $b\in Z^m$, some matrix $A\in Z^{m\times n}$ and some real symmetric matrix $F\in R^{n\times n}$. We show that $P$ can be formulated as a MAX-CUT problem whose quadratic form criterion is explicit from the data of … Read more
Constraint reduction is an essential method because the computational cost of the interior point methods can be effectively saved. Park and O’Leary proposed a constraint-reduced predictor-corrector algorithm for semidefinite programming with polynomial global convergence, but they did not show its superlinear convergence. We first develop a constraint-reduced algorithm for semidefinite programming having both polynomial global … Read more