We consider risk-averse formulations of stochastic linear programs having a structure that is common in real-life applications. Specifically, the optimization problem corresponds to controlling over a certain horizon a system whose dynamics is given by a transition equation depending affinely on an interstage dependent stochastic process. We put in place a rolling-horizon time consistent policy. … Read more
We consider the problems of finding a maximum clique in a graph and finding a maximum-edge biclique in a bipartite graph. Both problems are NP-hard. We write both problems as matrix-rank minimization and then relax them using the nuclear norm. This technique, which may be regarded as a generalization of compressive sensing, has recently been … Read more
We consider linear optimization over a fixed compact convex feasible region that is semi-algebraic (or, more generally, “tame”). Generically, we prove that the optimal solution is unique and lies on a unique manifold, around which the feasible region is “partly smooth”, ensuring finite identification of the manifold by many optimization algorithms. Furthermore, second-order optimality conditions … Read more
In this paper we propose and analyze a variant of the level method [4], which is an algorithm for minimizing nonsmooth convex functions. The main work per iteration is spent on 1) minimizing a piecewise-linear model of the objective function and on 2) projecting onto the intersection of the feasible region and a polyhedron arising … Read more
We propose a randomized method for general convex optimization problems; namely, the minimization of a linear function over a convex body. The idea is to generate N random points inside the body, choose the best one and cut the part of the body defined by the linear constraint. We first analyze the convergence properties of … Read more
The linearly constrained matrix rank minimization problem is widely applicable in many fields such as control, signal processing and system identification. The linearly constrained nuclear norm minimization is a convex relaxation of this problem. Although it can be cast as a semidefinite programming problem, the nuclear norm minimization problem is expensive to solve when the … Read more
Approximate proximal point algorithms (abbreviated as APPAs) are classical approaches for convex optimization problems and monotone variational inequalities. To solve the subproblems of these algorithms, the projection method takes the iteration in form of $u^{k+1} = P_{\Omega}[u^k-\alpha_k d^k]$. Interestingly, many of them can be paired such that $%\exists \tilde{u}^k, \tilde{u}^k = P_{\Omega}[u^k – \beta_kF(v^k)] = … Read more
This paper proposes an infeasible interior-point algorithm with full-Newton step for linear programming, which is an extension of the work of Roos (SIAM J. Optim., 16(4):1110–1136, 2006). We introduce a kernel function in the algorithm. For $p\in[0,1)$, the polynomial complexity can be proved and the result coincides with the best result for infeasible interior-point methods, … Read more
This paper describes the first phase of a project attempting to construct an efficient general-purpose nonlinear optimizer using an augmented Lagrangian outer loop with a relative error criterion, and an inner loop employing a state-of-the art conjugate gradient solver. The outer loop can also employ double regularized proximal kernels, a fairly recent theoretical development that … Read more
Image restoration models based on total variation (TV) have become popular since their introduction by Rudin, Osher, and Fatemi (ROF) in 1992. The dual formulation of this model has a quadratic objective with separable constraints, making projections onto the feasible set easy to compute. This paper proposes application of gradient projection (GP) algorithms to the … Read more