Let K be a closed convex cone with dual K-star in a finite-dimensional real Hilbert space V. A positive operator on K is a linear operator L on V such that L(K) is a subset of K. Positive operators generalize the nonnegative matrices and are essential to the Perron-Frobenius theory. We say that L is … Read more
Feasibility pump (FP) is a successful primal heuristic for mixed-integer linear programs (MILP). The algorithm consists of three main components: rounding fractional solution to a mixed-integer one, projection of infeasible solutions to the LP relaxation, and a randomization step used when the algorithm stalls. While many generalizations and improvements to the original Feasibility Pump have … Read more
This paper studies the matrix completion problem under arbitrary sampling schemes. We propose a new estimator incorporating both max-norm and nuclear-norm regularization, based on which we can conduct efficient low-rank matrix recovery using a random subset of entries observed with additive noise under general non-uniform and unknown sampling distributions. This method significantly relaxes the uniform … Read more
Given a polytope X, a monotone concave univariate function g, and two vectors c and d, we study the discrete optimization problem of finding a vertex of X that maximizes the utility function c’x + g(d’x). This problem has numerous applications in combinatorial optimization with a probabilistic objective, including estimation of project duration with stochastic … Read more
Adaptive robust optimization problems are usually solved approximately by restricting the adaptive decisions to simple parametric decision rules. However, the corresponding approximation error can be substantial. In this paper we show that two-stage robust and distributionally robust linear programs can often be reformulated exactly as conic programs that scale polynomially with the problem dimensions. Specifically, … Read more
We study two-stage adjustable robust linear programming in which the right-hand sides are uncertain and belong to a convex, compact uncertainty set. This problem is NP-hard, and the affine policy is a popular, tractable approximation. We prove that under standard and simple conditions, the two-stage problem can be reformulated as a copositive optimization problem, which … Read more
A formulation for the automated generation of algorithms via mathematical programming (optimization) is proposed. The formulation is based on the concept of optimizing within a parameterized family of algorithms, or equivalently a family of functions describing the algorithmic steps. The optimization variables are the parameters – within this family of algorithms- that encode algorithm design: … Read more
The minimization of a convex quadratic function under bound constraints is a fundamental building block for more complicated optimization problems. The active-set method introduced by [M. Bergounioux, K. Ito, and K. Kunisch. Primal-Dual Strategy for Constrained Optimal Control Problems. SIAM Journal on Control and Optimization, 37:1176–1194, 1999.] and [M. Bergounioux, M. Haddou, M. Hintermüller, and … Read more
A primal feasible active set method is presented for finding the unique solution of a Linear Complementarity Problem (LCP) with a P-matrix, which extends the globally convergent active set method for strictly convex quadratic problems with simple bounds proposed by [P. Hungerlaender and F. Rendl. A feasible active set method for strictly convex problems with … Read more
Quasi-Newton acceleration is an interesting tool to improve the performance of numerical methods based on the fixed-point paradigm. In this work the quasi-Newton technique will be applied to an inverse problem that comes from Positron Emission Tomography, whose fixed-point counterpart has been introduced recently. It will be shown that the improvement caused by the quasi-Newton … Read more