MPC

MPC 2017, ISSUE 4



Mathematical Programming Computation, Volume 9, Issue 4, December 2017

A branch-and-bound algorithm for instrumental variable quantile regression

Guanglin Xu, Samuel Burer

This paper studies a statistical problem called instrumental variable quantile regression (IVQR). We model IVQR as a convex quadratic program with complementarity constraints and—although this type of program is generally NPhard—we develop a branch-and-bound algorithm to solve it globally. We also derive bounds on key variables in the problem, which are valid asymptotically for increasing sample size. We compare our method with two well known global solvers, one of which requires the computed bounds. On random instances, our algorithm performs well in terms of both speed and robustness.

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Mathematical Programming Computation, Volume 9, Issue 4, December 2017

Lift-and-project cuts for convex mixed integer nonlinear programs

Mustafa R. Kilinc, Jeff Linderoth, James Luedtke

We describe a computationally effective method for generating lift-and-project cuts for convex mixed-integer nonlinear programs (MINLPs). The method relies on solving a sequence of cut-generating linear programs and in the limit generates an inequality as strong as the lift-and-project cut that can be obtained from solving a cut-generating nonlinear program. Using this procedure, we are able to approximately optimize over the rank one lift-and-project closure for a variety of convex MINLP instances. The results indicate that lift-and-project cuts have the potential to close a significant portion of the integrality gap for convex MINLPs. In addition, we find that using this procedure within a branch-and-cut solver for convex MINLPs significantly reduces the total solution time for many instances.We also demonstrate that combining lift-and-project cuts with an extended formulation that exploits separability of convex functions yields significant improvements in both relaxation bounds and the time to calculate the relaxation. Overall, these results suggest that with an effective separation routine, like the one proposed here, lift-and-project cuts may be as effective for solving convex MINLPs as they have been for solving mixed-integer linear programs.

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Mathematical Programming Computation, Volume 9, Issue 4, December 2017

Lifted collocation integrators for direct optimal control in ACADO toolkit

Rien Quirynen, Sebastien Gros, Boris Houska, Moritz Diehl

This paper presents a class of efficient Newton-type algorithms for solving the nonlinear programs (NLPs) arising from applying a direct collocation approach to continuous time optimal control. The idea is based on an implicit lifting technique including a condensing and expansion step, such that the structure of each subproblem corresponds to that of themultiple shooting method for direct optimal control.We establish the mathematical equivalence between the Newton iteration based on direct collocation and the proposed approach, and we discuss the computational advantages of a lifted collocation integrator. In addition, we investigate different inexact versions of the proposed scheme and study their convergence and computational properties. The presented algorithms are implemented as part of the open-source ACADO code generation software for embedded optimization. Their performance is illustrated on a benchmark case study of the optimal control for a chain of masses. Based on these results, the use of lifted collocationwithin direct multiple shooting allows for a computational speedup factor of about 10 compared to a standard collocation integrator and a factor in the range of 10–50 compared to direct collocation using a general-purpose sparse NLP solver.

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Mathematical Programming Computation, Volume 9, Issue 4, December 2017

On the computational efficiency of subgradient methods: a case study with Lagrangian bounds

Antonio Frangioni, Bernard Gendron, Enrico Gorgone

Subgradient methods (SM) have long been the preferred way to solve the large-scale Nondifferentiable Optimization problems arising from the solution of Lagrangian Duals (LD) of Integer Programs (IP). Although other methods can have better convergence rate in practice, SM have certain advantages that may make them competitive under the right conditions. Furthermore, SMhave significantly progressed in recent years, and new versions have been proposed with better theoretical and practical performances in some applications.We computationally evaluate a large class of SM in order to assess if these improvements carry over to the IP setting. For this we build a unified scheme that covers many of the SMproposed in the literature, comprised some often overlooked features like projection and dynamic generation of variables. We fine-tune the many algorithmic parameters of the resulting large class of SM, and we test them on two different LDs of the Fixed-Charge Multicommodity Capacitated Network Design problem, in order to assess the impact of the characteristics of the problem on the optimal algorithmic choices. Our results show that, if extensive tuning is performed, SM can be competitive with more sophisticated approaches when the tolerance required for solution is not too tight, which is the case when solving LDs of IPs.

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MPC 2017, ISSUE 3



Mathematical Programming Computation, Volume 9, Issue 3, September 2017

Convex quadratic relaxations for mixed-integer nonlinear programs in power systems

Hassan Hijazi, Carleton Coffrin, Pascal Van Hentenryck

This paper presents a set of new convex quadratic relaxations for nonlinear and mixed-integer nonlinear programs arising in power systems. The considered models are motivated by hybrid discrete/continuous applications where existing approximations do not provide optimality guarantees. The new relaxations offer computational efficiency along with minimal optimality gaps, providing an interesting alternative to state-of-the-art semidefinite programming relaxations. Three case studies in optimal power flow, optimal transmission switching and capacitor placement demonstrate the benefits of the new relaxations.

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Mathematical Programming Computation, Volume 9, Issue 3, September 2017

Extended formulations in mixed integer conic quadratic programming

Juan Pablo Vielma, Iain Dunning, Joey Huchette, Miles Lubin

In this paper we consider the use of extended formulations in LP-based algorithms for mixed integer conic quadratic programming (MICQP). Extended formulations have been used by Vielma et al. (INFORMS J Comput 20: 438–450, 2008) and Hijazi et al. (Comput Optim Appl 52: 537–558, 2012) to construct algorithms for MICQP that can provide a significant computational advantage. The first approach is based on an extended or lifted polyhedral relaxation of the Lorentz cone by Ben-Tal and Nemirovski (Math Oper Res 26(2): 193–205 2001) that is extremely economical, but whose approximation quality cannot be iteratively improved. The second is based on a lifted polyhedral relaxation of the euclidean ball that can be constructed using techniques introduced by Tawarmalani and Sahinidis (Math Programm 103(2): 225–249, 2005). This relaxation is less economical, but its approximation quality can be iteratively improved. Unfortunately, while the approach of Vielma, Ahmed and Nemhauser is applicable for general MICQP problems, the approach of Hijazi, Bonami and Ouorou can only be used for MICQP problems with convex quadratic constraints. In this paper we show how a homogenization procedure can be combined with the technique by Tawarmalani and Sahinidis to adapt the extended formulation used by Hijazi, Bonami and Ouorou to a class of conic mixed integer programming problems that include general MICQP problems. We then compare the effectiveness of this new extended formulation against traditional and extended formulation-based algorithms forMICQP.We find that this new formulation can be used to improve various LP-based algorithms. In particular, the formulation provides an easy-to-implement procedure that, in our benchmarks, significantly improved the performance of commercial MICQP solvers.

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Mathematical Programming Computation, Volume 9, Issue 3, September 2017

New computer-based search strategies for extreme functions of the Gomory–Johnson infinite group problem

Matthias Köppe, Yuan Zhou

We describe new computer-based search strategies for extreme functions for the Gomory–Johnson infinite group problem. They lead to the discovery of new extreme functions, whose existence settles several open questions.

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MPC 2017, ISSUE 2



Mathematical Programming Computation, Volume 9, Issue 2, June 2017

Dijkstra meets Steiner: a fast exact goal-oriented Steiner tree algorithm

Stefan Hougardy, Jannik Silvanus, Jens Vygen

We present a new exact algorithm for the Steiner tree problem in edge-weighted graphs. Our algorithm improves the classical dynamic programming approach by Dreyfus and Wagner. We achieve a significantly better practical performance via pruning and future costs, a generalization of a well-known concept to speed up shortest path computations. Our algorithm matches the best known worst-case run time and has a fast, often superior, practical performance: on some large instances originating from VLSI design, previous best run times are improved upon by orders of magnitudes. We are also able to solve larger instances of the d-dimensional rectilinear Steiner tree problem for d?{3,4,5}d?{3,4,5}, whose Hanan grids contain up to several millions of edges.

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Mathematical Programming Computation, Volume 9, Issue 2, June 2017

Thinning out Steiner trees: a node-based model for uniform edge costs

Matteo Fischetti, Markus Leitner, Ivana Ljubic, Martin Luipersbeck, Michele Monaci, Max Resch, Domenico Salvagnin, Markus Sinnl

The Steiner tree problem is a challenging NP-hard problem. Many hard instances of this problem are publicly available, that are still unsolved by state-of-the-art branch-and-cut codes. A typical strategy to attack these instances is to enrich the polyhedral description of the problem, and/or to implement more and more sophisticated separation procedures and branching strategies. In this paper we investigate the opposite viewpoint, and try to make the solution method as simple as possible while working on the modeling side. Our working hypothesis is that the extreme hardness of some classes of instances mainly comes from over-modeling, and that some instances can become quite easy to solve when a simpler model is considered. In other words, we aim at “thinning out” the usual models for the sake of getting a more agile framework. In particular, we focus on a model that only involves node variables, which is rather appealing for the “uniform” cases where all edges have the same cost. In our computational study, we first show that this new model allows one to quickly produce very good (sometimes proven optimal) solutions for notoriously hard instances from the literature. In some cases, our approach takes just few seconds to prove optimality for instances never solved (even after days of computation) by the standard methods. Moreover, we report improved solutions for several SteinLib instances, including the (in)famous hypercube ones. We also demonstrate how to build a unified solver on top of the new node-based model and the previous state-of-the-art model (defined in the space of arc and node variables). The solver relies on local branching, initialization heuristics, preprocessing and local search procedures. A filtering mechanism is applied to automatically select the best algorithmic ingredients for each instance individually. The presented solver is the winner of the DIMACS Challenge on Steiner trees in most of the considered categories.

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Mathematical Programming Computation, Volume 9, Issue 2, June 2017

SCIP-Jack—a solver for STP and variants with parallelization extensions

Gerald Gamrath, Thorsten Koch, Stephen J. Maher, Daniel Rehfeldt, Yuji Shinano

The Steiner tree problem in graphs is a classical problem that commonly arises in practical applications as one of many variants. While often a strong relationship between different Steiner tree problem variants can be observed, solution approaches employed so far have been prevalently problem-specific. In contrast, this paper introduces a general-purpose solver that can be used to solve both the classical Steiner tree problem and many of its variants without modification. This versatility is achieved by transforming various problem variants into a general form and solving them by using a state-of-the-art MIP-framework. The result is a high-performance solver that can be employed in massively parallel environments and is capable of solving previously unsolved instances.

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Mathematical Programming Computation, Volume 9, Issue 2, June 2017

Swap-vertex based neighborhood for Steiner tree problems

Zhang-Hua Fu, Jin-Kao Hao

Steiner tree problems (STPs) are very important in both theory and practice. In this paper, we introduce a powerful swap-vertex move operator which can be used as a basic element of any neighborhood search heuristic to solve many STP variants. Given the incumbent solution tree T, the swap-vertex move operator exchanges a vertex in T with another vertex out of T, and then attempts to construct a minimum spanning tree, leading to a neighboring solution (if feasible). We develop a series of dynamic data structures, which allow us to efficiently evaluate the feasibility of swap-vertex moves. Additionally, in order to discriminate different swap-vertex moves corresponding to the same objective value, we also develop an auxiliary evaluation function. We present a computational assessment based on a number of challenging problem instances (corresponding to three representative STP variants) which clearly shows the effectiveness of the techniques introduced in this paper. Particularly, as a key element of our KTS algorithm which participated in the 11th DIMACS implementation challenge, the swap-vertex operator as well as the auxiliary evaluation function contributed significantly to the excellent performance of our algorithm.

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MPC 2017, ISSUE 1



Mathematical Programming Computation, Volume 9, Issue 1, March 2017

Computing convex hulls and counting integer points with polymake

Benjamin Assarf, Ewgenij Gawrilow, Katrin Herr, Michael Joswig, Benjamin Lorenz, Andreas Paffenholz, Thomas Rehn

The main purpose of this paper is to report on the state of the art of computing integer hulls and their facets as well as counting lattice points in convex polytopes. Using the polymake system we explore various algorithms and implementations. Our experience in this area is summarized in ten “rules of thumb”.

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Mathematical Programming Computation, Volume 9, Issue 1, March 2017

Fast separation for the three-index assignment problem

Trivikram Dokka, Ioannis Mourtos, Frits C. R. Spieksma

A critical step in a cutting plane algorithm is separation, i.e., establishing whether a given vector x violates an inequality belonging to a specific class. It is customary to express the time complexity of a separation algorithm in the number of variables n. Here, we argue that a separation algorithm may instead process the vector containing the positive components of x, denoted as supp(x), which offers a more compact representation, especially if x is sparse; we also propose to express the time complexity in terms of |supp(x)|. Although several well-known separation algorithms exploit the sparsity of x, we revisit this idea in order to take sparsity explicitly into account in the time-complexity of separation and also design faster algorithms. We apply this approach to two classes of facet-defining inequalities for the three-index assignment problem, and obtain separation algorithms whose time complexity is linear in |supp(x)| instead of n. We indicate that this can be generalized to the axial k-index assignment problem and we show empirically how the separation algorithms exploiting sparsity improve on existing ones by running them on the largest instances reported in the literature.

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Mathematical Programming Computation, Volume 9, Issue 1, March 2017

Improved branch-cut-and-price for capacitated vehicle routing

Diego Pecin, Artur Pessoa, Marcus Poggi, Eduardo Uchoa

The best performing exact algorithms for the capacitated vehicle routing problem developed in the last 10 years are based in the combination of cut and column generation. Some authors only used cuts expressed over the variables of the original formulation, in order to keep the pricing subproblem relatively easy. Other authors could reduce the duality gaps by also using a restricted number of cuts over the master LP variables, stopping when the pricing becomes prohibitively hard. A particularly effective family of such cuts are the subset row cuts. This work introduces a technique for greatly reducing the impact on the pricing of these cuts, thus allowing much more cuts to be added. The newly proposed branch-cut-and-price algorithm also incorporates and combines for the first time (often in an improved way) several elements found in previous works, like route enumeration and strong branching. All the instances used for benchmarking exact algorithms, with up to 199 customers, were solved to optimality. Moreover, some larger instances with up to 360 customers, only considered before by heuristic methods, were solved too.

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Mathematical Programming Computation, Volume 9, Issue 1, March 2017

On efficiently combining limited-memory and trust-region techniques

Oleg Burdakov, Lujin Gong, Spartak Zikrin, Ya-xiang Yuan

Limited-memory quasi-Newton methods and trust-region methods represent two efficient approaches used for solving unconstrained optimization problems. A straightforward combination of them deteriorates the efficiency of the former approach, especially in the case of large-scale problems. For this reason, the limited-memory methods are usually combined with a line search. We show how to efficiently combine limited-memory and trust-region techniques. One of our approaches is based on the eigenvalue decomposition of the limited-memory quasi-Newton approximation of the Hessian matrix. The decomposition allows for finding a nearly-exact solution to the trust-region subproblem defined by the Euclidean norm with an insignificant computational overhead as compared with the cost of computing the quasi-Newton direction in line-search limited-memory methods. The other approach is based on two new eigenvalue-based norms. The advantage of the new norms is that the trust-region subproblem is separable and each of the smaller subproblems is easy to solve. We show that our eigenvalue-based limited-memory trust-region methods are globally convergent. Moreover, we propose improved versions of the existing limited-memory trust-region algorithms. The presented results of numerical experiments demonstrate the efficiency of our approach which is competitive with line-search versions of the L-BFGS method.

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