Adding cuts based on copositive matrices, we propose to improve Lovász’ bound on the clique number and its tightening introduced by McEliece, Rodemich, Rumsey, and Schrijver. Candidates for cheap and efficient copositivity cuts of this type are obtained from graphs with known clique number. The cost of previously established semidefinite programming bound hierarchies rapidly increases … Read more
An SDP relaxation based method is developed to solve the localization problem in sensor networks using incomplete and inaccurate distance information. The problem is set up to find a set of sensor positions such that given distance constraints are satisfied. The nonconvex constraints in the formulation are then relaxed in order to yield a semidefinite … Read more
We analyze the semidefinite programming (SDP) based model and method for the position estimation problem in sensor network localization and other Euclidean distance geometry applications. We use SDP duality and interior–point algorithm theories to prove that the SDP localizes any network or graph that has unique sensor positions to fit given distance measures. Therefore, we … Read more
We consider the problem of computing the outer-radii of point sets. In this problem, we are given integers $n, d, k$ where $k \le d$, and a set $P$ of $n$ points in $R^d$. The goal is to compute the {\em outer $k$-radius} of $P$, denoted by $\kflatr(P)$, which is the minimum, over all $(d-k)$-dimensional … Read more
The convergence of primal and dual central paths associated to entropy and exponential functions, respectively, for semidefinite programming problem are studied in this paper. As an application, the proximal point method with the Kullback-Leibler distance applied to semidefinite programming problems is considered, and the convergence of primal and dual sequences is proved. Citation Journal of … Read more
Semidefinite programming has been an interesting and active area of research for several years. In semidefinite programming one optimizes a convex (often linear) objective function subject to a system of linear matrix inequality constraints. Despite its numerous applications, algorithms for solving semidefinite programming problems are restricted to problems of moderate size because the computation time … Read more
A notorious open problem in the field of rendezvous search is to decide the rendezvous value of the symmetric rendezvous search problem on the line, when the initial distance apart between the two players is 2. We show that the symmetric rendezvous value is within the interval (4.1520, 4.2574), which considerably improves the previous best … Read more
We present a target-following framework for semidefinite programming, which generalizes the target-following framework for linear programming. We use this framework to build weighted path-following interior-point algorithms of three distinct flavors: short-step, predictor-corrector, and large-update. These algorithms have worse-case iteration bounds that parallel their counterparts in linear programming. We further consider the problem of finding analytic … Read more
Exploiting sparsity is essential to improve the efficiency of solving large optimization problems. We present a method for recognizing the underlying sparsity structure of a nonlinear partially separable problem, and show how the sparsity of the Hessian matrices of the problem’s functions can be improved by performing a nonsingular linear transformation in the space corresponding … Read more
Korkin and Zolotarev showed that if $$\sum_i A_i(x_i-\sum_{j>i} \alpha_{ij}x_j)^2$$ is the Lagrange expansion of a Korkin–Zolotarev reduced positive definite quadratic form, then $A_{i+1}\geq \frac{3}{4} A_i$ and $A_{i+2}\geq \frac{2}{3}A_i$. They showed that the implied bound $A_{5}\geq \frac{4}{9}A_1$ is not attained by any KZ-reduced form. We propose a method to optimize numerically over the set of Lagrange … Read more