The use of convex optimization for the recovery of sparse signals from incomplete or compressed data is now common practice. Motivated by the success of basis pursuit in recovering sparse vectors, new formulations have been proposed that take advantage of different types of sparsity. In this paper we propose an efficient algorithm for solving a … Read more
Inspired by the One Laptop Per Child project, we consider mesh networks that connect devices that cannot recharge their batteries easily. We study how the mesh should retransmit information to make use of the energy stored in each of the nodes effectively. The solution that minimizes the total energy spent by the whole network may … Read more
The nuclear norm is widely used to induce low-rank solutions for many optimization problems with matrix variables. Recently, it has been shown that the augmented Lagrangian method (ALM) and the alternating direction method (ADM) are very efficient for many convex programming problems arising from various applications, provided that the resulting subproblems are sufficiently simple to … Read more
We study the problem of detecting a maximum embedded network submatrix in a {-1,0,+1}-matrix. Our aim is to solve the problem to optimality. We introduce a 0-1 integer linear formulation for this problem based on its representation over a signed graph. A polyhedral study is presented and a branch-and-cut algorithm is described for finding an … Read more
We consider optimization of nonlinear objective functions that balance $d$ linear criteria over $n$-element independence systems presented by linear-optimization oracles. For $d=1$, we have previously shown that an $r$-best approximate solution can be found in polynomial time. Here, using an extended Erdos-Ko-Rado theorem of Frankl, we show that for $d=2$, finding a $\rho n$-best solution … Read more
The SDPA (SemiDefinite Programming Algorithm) Project launched in 1995 has been known to provide high-performance packages for solving large-scale Semidefinite Programs (SDPs). SDPA Ver. 6 solves efficiently large-scale dense SDPs, however, it required much computation time compared with other software packages, especially when the Schur complement matrix is sparse. SDPA Ver. 7 is now completely … Read more
We present an efficient decomposition algorithm for single-stage, stochastic linear programs, where conditional value at risk (CVaR) appears as a risk measure in multiple constraints. It starts with a well-known nonlinear, convex reformulation of conditional value at risk constraints, and establishes the connection to a combinatorially large polyhedral representation of the convex feasible set induced … Read more
We consider the synthesis problem of Compressed Sensing: given $s$ and an $M\times n$ matrix $A$, extract from $A$ an $m\times n$ submatrix $A_m$, with $m$ as small as possible, which is $s$-good, that is, every signal $x$ with at most $s$ nonzero entries can be recovered from observation $A_m x$ by $\ell_1$ minimization: $x … Read more
Interior point method (IPM) defines a search direction at each interior point of a region. These search directions form a direction field which in turn gives rise to a system of ordinary differential equations (ODEs). The solutions of the system of ODEs can be viewed as underlying paths in the interior of the region. In … Read more
In this paper we propose new methods for solving huge-scale optimization problems. For problems of this size, even the simplest full-dimensional vector operations are very expensive. Hence, we propose to apply an optimization technique based on random partial update of decision variables. For these methods, we prove the global estimates for the rate of convergence. … Read more