Cardinality Minimization, Constraints, and Regularization: A Survey

We survey optimization problems that involve the cardinality of variable vectors in constraints or the objective function. We provide a unified viewpoint on the general problem classes and models, and give concrete examples from diverse application fields such as signal and image processing, portfolio selection, or machine learning. The paper discusses general-purpose modeling techniques and … Read more

Minimizing Airplane Boarding Time

The time it takes passengers to board an airplane is known to influence the turn-around time of the aircraft and thus bears a significant cost-saving potential for airlines. Although minimizing boarding time therefore is the most important goal from an economic perspective, previous efforts to design efficient boarding strategies apparently never tackled this task directly. … Read more

Computing the Spark: Mixed-Integer Programming for the (Vector) Matroid Girth Problem

We investigate the NP-hard problem of computing the spark of a matrix (i.e., the smallest number of linearly dependent columns), a key parameter in compressed sensing and sparse signal recovery. To that end, we identify polynomially solvable special cases, gather upper and lower bounding procedures, and propose several exact (mixed-)integer programming models and linear programming … Read more

A Primal-Dual Homotopy Algorithm for l_1-Minimization with l_inf-Constraints

In this paper we propose a primal-dual homotopy method for $\ell_1$-minimization problems with infinity norm constraints in the context of sparse reconstruction. The natural homotopy parameter is the value of the bound for the constraints and we show that there exists a piecewise linear solution path with finitely many break points for the primal problem … Read more

Solving Basis Pursuit: Heuristic Optimality Check and Solver Comparison

The problem of finding a minimum l^1-norm solution to an underdetermined linear system is an important problem in compressed sensing, where it is also known as basis pursuit. We propose a heuristic optimality check as a general tool for l^1-minimization, which often allows for early termination by “guessing” a primal-dual optimal pair based on an … Read more

An Infeasible-Point Subgradient Method Using Adaptive Approximate Projections

We propose a new subgradient method for the minimization of convex functions over a convex set. Common subgradient algorithms require an exact projection onto the feasible region in every iteration, which can be efficient only for problems that admit a fast projection. In our method we use inexact adaptive projections requiring to move within a … Read more