Hybrid extragradient proximal algorithm coupled with parametric approximation and penalty/barrier methods

In this paper we study the hybrid extragradient method coupled with approximation and penalty schemes for minimization problems. Under certain hypotheses, that include for example the case of Tikhonov regularization, we prove convergence of the method to the solution set of our minimization problem. When we use schemes of penalization or barrier we can show … Read more

Nonlinear Matroid Optimization and Experimental Design

We study the problem of optimizing nonlinear objective functions over matroids presented by oracles or explicitly. Such functions can be interpreted as the balancing of multi-criteria optimization. We provide a combinatorial polynomial time algorithm for arbitrary oracle-presented matroids, that makes repeated use of matroid intersection, and an algebraic algorithm for vectorial matroids. Our work is … Read more

A Fast Algorithm For Image Deblurring with Total Variation Regularization

We propose and test a simple algorithmic framework for recovering images from blurry and noisy observations based on total variation (TV) regularization when a blurring point-spread function is given. Using a splitting technique, we construct an iterative procedure of alternately solving a pair of easy subproblems associated with an increasing sequence of penalty parameter values. … Read more

New stopping criteria for detecting infeasibility in conic optimization

Detecting infeasibility in conic optimization and providing certificates for infeasibility pose a bigger challenge than in the linear case due to the lack of strong duality. In this paper we generalize the approximate Farkas lemma of Todd and Ye from the linear to the general conic setting, and use it to propose stopping criteria for … Read more

Stopping Rules for Box-Constrained Stochastic Global Optimization

We present three new stopping rules for Multistart based methods. The first uses a device that enables the determination of the coverage of the bounded search domain. The second is based on the comparison of asymptotic expectation values of observable quantities to the actually measured ones. The third offers a probabilistic estimate for the number … Read more

Approximate Solutions for Deterministic and Stochastic Multi-Dimensional Sequencing

We investigate the problem of sequencing jobs that have multiple components. Each component of the job needs to be processed independently on a specified machine. We derive approximate algorithms for the problem of scheduling such vector jobs to minimize their total completion time in the deterministic as well as stochastic setting. In particular, we propose … Read more

A Fixed-Point Continuation Method for l_1-Regularized Minimization with Applications to Compressed Sensing

We consider solving minimization problems with $\ell_1$-regularization: $$\min \|x\|_1 + \mu f(x),$$ particularly for $f(x) = \frac{1}{2}\|Ax-b\|_M^2$ where $A \in \R^{m \times n}$ with $m < n$. Our goal is to construct efficient and robust algorithms for solving large-scale problems with dense data, and our approach is based on two powerful algorithmic ideas, operator-splitting and ... Read more

A 2-BFGS updating in a trust region framework

We present a new matrix-free method for the trust region subproblem, assuming that the approximate Hessian is updated by the limited memory BFGS formula with m = 2. The resulting updating scheme, called 2-BFGS, give us the ability to determine via simple formulas the eigenvalues of the resulting approximation. Thus, at each iteration, we can … Read more

ASTRAL: An Active Set \inftyhBcTrust-Region Algorithm for Box Constrained Optimization

An algorithm for solving large-scale nonlinear optimization problems with simple bounds is described. The algorithm is an $\ell_\infty$-norm trust-region method that uses both active set identification techniques as well as limited memory BFGS updating for the Hessian approximation. The trust-region subproblems are solved using primal-dual interior point techniques that exploit the structure of the limited … Read more

Pareto Optima of Multicriteria Integer Linear Programs

We settle the computational complexity of fundamental questions related to multicriteria integer linear programs, when the dimensions of the strategy space and of the outcome space are considered fixed constants. In particular we construct: 1. polynomial-time algorithms to exactly determine the number of Pareto optima and Pareto strategies; 2. a polynomial-space polynomial-delay prescribed-order enumeration algorithm … Read more