Portfolio optimization problems involving Value-at-Risk (VaR) are often computationally intractable and require complete information about the return distribution of the portfolio constituents, which is rarely available in practice. These difficulties are compounded when the portfolio contains derivatives. We develop two tractable conservative approximations for the VaR of a derivative portfolio by evaluating the worst-case VaR … Read more
In this work, we propose an inexact interior proximal type algorithm for solving convex second-order cone programs. This kind of problems consists of minimizing a convex function (possibly nonsmooth) over the intersection of an affine linear space with the Cartesian product of second-order cones. The proposed algorithm uses a distance variable metric, which is induced … Read more
Robust portfolio optimization finds the worst-case portfolio return given that the asset returns are realized within a prescribed uncertainty set. If the uncertainty set is not too large, the resulting portfolio performs well under normal market conditions. However, its performance may substantially degrade in the presence of market crashes, that is, if the asset returns … Read more
Given a 0-1 integer programming problem, several authors have introduced sequential relaxation techniques — based on linear and/or semidefinite programming — that generate the convex hull of integer points in at most $n$ steps. In this paper, we introduce a sequential relaxation technique, which is based on $p$-order cone programming ($1 \le p \le \infty$). … Read more
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
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
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. Citation AdvOl-Report#2004/15 McMaster University, Advanced Optimization Laboratory Article Download View The Q Method for Second-order Cone Programming
We propose a pivotting procedure for a class of Second-Order Cone Programming (SOCP) having one second-order cone. We introduce a dictionary, basic variables, nonbasic variables, and other necessary notions to define a pivot for the class of SOCP. In a pivot, two-dimensional SOCP subproblems are solved to decide which variables should be entering to or … Read more
We consider primal-dual algorithms for certain types of infinite-dimensional optimization problems. Our approach is based on the generalization of the technique of finite-dimensional Euclidean Jordan algebras to the case of infinite-dimensional JB-algebras of finite rank. This generalization enables us to develop polynomial-time primal-dual algorithms for “infinite-dimensional second-order cone programs.” We consider as an example a … Read more