Strong formulations for quadratic optimization with M-matrices and indicator variables
We study quadratic optimization with indicator variables and an M-matrix, i.e., a PSD matrix with non-positive off-diagonal entries, which arises directly in image segmentation and portfolio optimization with transaction costs, as well as a substructure of general quadratic optimization problems. We prove, under mild assumptions, that the minimization problem is solvable in polynomial time by showing its equivalence to a submodular minimization problem. To strengthen the formulation, we decompose the quadratic function into a sum of simple quadratic functions with at most two indicator variables each and provide the convex-hull descriptions of these sets. We also describe strong conic quadratic valid inequalities. Preliminary computational experiments indicate that the proposed inequalities can substantially improve the strength of the continuous relaxations with respect to the standard perspective reformulation.Atamtürk, A. and Gómez, A..
Simplex QP-based methods for minimizing a conic quadratic objective over polyehdra
We consider minimizing a conic quadratic objective over a polyhedron. Such problems arise in parametric value-at-risk minimization, portfolio optimization, and robust optimization with ellipsoidal objective uncertainty; and they can be solved by polynomial interior point algorithms for conic quadratic optimization. However, interior point algorithms are not well-suited for branch-and-bound algorithms for the discrete counterparts of these problems due to the lack of effective warm starts necessary for the efficient solution of convex relaxations repeatedly at the nodes of the search tree. In order to overcome this shortcoming, we reformulate the problem using the perspective of its objective. The perspective reformulation lends itself to simple coordinate descent and bisection algorithms utilizing the simplex method for quadratic programming, which makes the solution methods amenable to warm starts and suitable for branch-and-bound algorithms. We test the simplex-based quadratic programming algorithms to solve convex as well as discrete instances and compare them with the state-of-the-art approaches. The computational experiments indicate that the proposed algorithms scale much better than interior point algorithms and return higher precision solutions. In our experiments, for large convex instances, they provide up to 22x speed-up. For smaller discrete instances, the speed-up is about 13x over a barrier-based branch-and-bound algorithm and 6x over the LP-based branch-and-bound algorithm with extended formulations.
Atamtürk, A. and Gómez, A..
Submodularity in conic quadratic mixed 0-1 optimization
We describe strong convex valid inequalities for conic quadratic mixed 0-1 optimization. The inequalities exploit the submodularity of the binary restrictions and are based on the polymatroid inequalities over binaries for the diagonal case. We prove that the convex inequalities completely describe the convex hull of a single conic quadratic constraint as well as the rotated cone constraint over binary variables and unbounded continuous variables. We then generalize and strengthen the inequalities by incorporating additional constraints of the optimization problem. Computational experiments on mean-risk optimization with correlations, assortment optimization, and robust conic quadratic optimization indicate that the new inequalities strengthen the convex relaxations substantially and lead to significant performance improvements.
Atamtürk, A. and Gómez, A..
Maximizing a class of utility functions over the vertices of a polytope
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 times, in reliability models, in multinomial logit models and in robust optimization. We show that the problem is NP-hard for any strictly concave function g even for simple polytopes, such as the uniform matroid, assignment and path polytopes; and propose a 1/2-approximation algorithm for it. We discuss improvements for special cases where g is the square root, log utility, negative exponential utility and multinomial logit probability function. In particular, for the square root function, the approximation ratio is 4/5. We also propose a 1.25-approximation algorithm for a class of minimization problems in which the maximization of the utility function appears as a subproblem. Although the worst case bounds are tight, computational experiments indicate that the suggested approach finds solutions within 1-2% optimality gap for most of the instances, and can be considerably faster than the existing alternatives.
Atamtürk, A. and Gómez, A. (2016), Maximizing a class of utility functions over the vertices of a polytope. Operations Research 65:433-445.
Three-Partition Flow Cover Inequalities for Constant Capacity Fixed-Charge Network Flow Problems
Flow cover inequalities are among the most effective valid inequalities for capacitated fixed-charge network flow problems. These valid inequalities are based on implications for the flow quantity on the cut arcs of a two-partitioning of the network, depending on whether some of the cut arcs are open or closed. As the implications are only on the cut arcs, flow cover inequalities can be obtained by collapsing a subset of nodes into a single node. In this article, we derive new valid inequalities for the capacitated fixed-charge network flow problem by exploiting additional information from the network. In particular, the new inequalities are based on a three partitioning of the nodes. The new three-partition flow cover inequalities include the flow cover inequalities as a special case. We discuss the constant capacity case and give a polynomial separation algorithm for the inequalities. Finally, we report computational results with the new inequalities for networks with different characteristics.
Atamtürk, A., Gómez, A. and Küçükyavuz, S. (2016), Three-partition flow cover inequalities for constant capacity fixed-charge network flow problems. NETWORKS 67: 299–315.
On Modeling Stochastic Travel and Service Times in Vehicle Routing
Vehicle routing problems with stochastic travel and service times (VRPSTT) consist of designing transportation routes of minimal expected cost over a network where travel and service times are represented by random variables. Most of the existing approaches for VRPSTT are conceived to exploit the properties of the distributions assumed for the random variables. Therefore, these methods are tied to a given family of distributions and subject to strong modeling assumptions. We propose an alternative way to model travel and service times in VRPSTT without making many assumptions regarding such distributions. To illustrate our approach, we embed it into a state-of-the-art routing engine and use it to conduct experiments on instances with different travel and service time distributions
Gómez, A., Mariño, R., Akhavan-Tabatabaei, R., Medaglia, A. and Mendoza, J. (2016), On Modeling Stochastic Travel and Service Times in Vehicle Routing. Transportation Science 50:627-641.