We consider the problem of computing the smallest contact forces, with point-contact friction model, that can hold an object in equilibrium against a known external applied force and torque. It is known that the force optimization problem (FOP) can be formulated as a semidefinite programming problem (SDP), or a second-order cone problem (SOCP), and so … Read more
We describe a polynomial-size conic quadratic reformulation for a machine-job assignment problem with separable convex cost. Because the conic strengthening is based on only the objective of the problem, it can also be applied to other problems with similar cost functions. Computational results demonstrate the effectiveness of the conic reformulation. CitationAppeared in Operations Research Letters. … Read more
When is the linear image of a closed convex cone closed? We present very simple, and intuitive necessary conditions, which 1) unify, and generalize seemingly disparate, classical sufficient conditions: polyhedrality of the cone, and “Slater” type conditions; 2) are necessary and sufficient, when the dual cone belongs to a class, that we call nice cones. … Read more
When the mean vectors and the covariance matrices of two classes are available in a binary classification problem, Lanckriet et al.\ \cite{mpm} propose a minimax approach for finding a linear classifier which minimizes the worst-case (maximum) misclassification probability. We extend the minimax approach to a multiple classification problem, where the number $m$ of classes could … Read more
Stochastic programming, despite its immense modeling capabilities, is well known to be computationally excruciating. In this paper, we introduce a unified framework of approximating multiperiod stochastic programming from the perspective of robust optimization. Specifically, we propose a framework that integrates multistage modeling with safeguarding constraints. The framework is computationally tractable in the form of second … Read more
In this paper, we are concerned with nonlinear minimization problems with second order cone constraints. A primal-dual interior point method is proposed for solving the problems. We also propose a new primal-dual merit function by combining the barrier penalty function and the potential function within the framework of the line search strategy, and show the … Read more
In this paper, we show a way to exploit sparsity in the problem data in a primal-dual potential reduction method for solving a class of semidefinite programs. When the problem data is sparse, the dual variable is also sparse, but the primal one is not. To avoid working with the dense primal variable, we apply … Read more
We present a primal-dual interior-point algorithm for second-order conic optimization problems based on a specific class of kernel functions. This class has been investigated earlier for the case of linear optimization problems. In this paper we derive the complexity bounds $O(\sqrt{N})(\log N)\log\frac{N}{\epsilon})$ for large- and $O(\sqrt{N})\log\frac{N}{\epsilon}$ for small- update methods, respectively. Here $N$ denotes the … Read more
The second-order cone programming problem is reformulated into several new systems of nonlinear equations. Assume the perturbation of the data is in a certain neighborhood of zero. Then starting from a solution to the old problem, the semismooth Newton’s iterates converge Q-quadratically to a solution of the perturbed problem. The algorithm is globalized. Numerical examples … Read more
Based on the Q method for SDP, we develop the Q method for SOCP. A modified Q method is also introduced. Properties of the algorithms are discussed. Convergence proofs are given. Finally, we present numerical results. CitationAdvOl-Report#2004/15 McMaster University, Advanced Optimization LaboratoryArticleDownload View PDF