Commercial Mixed Integer Programming Solver Research Articles

Mediating governance goals with patients and nurses satisfaction: a multi-actor multi-objective problem including fairness

ABSTRACT Balancing conflicting goals among different stakeholders is a challenging problem in various application domains. In this paper, we analyse it in the context of home healthcare. Fairness objective functions for nurses and patients are combined with system-level governance goals of the territorial centers in charge of the assistance service. From the solution of several multi-objective problems including one objective for each actor hierarchically ordered, the best goal for each stakeholder is identified and interesting managerial conclusions are drawn. To efficiently solve large-size instances, we introduce a parallel Adaptive Large Neighborhood Search algorithm with destroy and repair operators customised to solve multi-objective multi-actor problems. The algorithm proves to be highly efficient and effective when compared to a commercial Mixed Integer Programming solver, either in its plain form or enforced by a mild start and a primal heuristic. Additionally, we devise a metaheuristic method to generate the Pareto frontier of an instance.

Domain-Independent Dynamic Programming: Generic State Space Search for Combinatorial Optimization

For combinatorial optimization problems, model-based approaches such as mixed-integer programming (MIP) and constraint programming (CP) aim to decouple modeling and solving a problem: the `holy grail' of declarative problem solving. We propose domain-independent dynamic programming (DIDP), a new model-based paradigm based on dynamic programming (DP). While DP is not new, it has typically been implemented as a problem-specific method. We propose Dynamic Programming Description Language (DyPDL), a formalism to define DP models, and develop Cost-Algebraic A* Solver for DyPDL (CAASDy), a generic solver for DyPDL using state space search. We formalize existing problem-specific DP and state space search methods for combinatorial optimization problems as DP models in DyPDL. Using CAASDy and commercial MIP and CP solvers, we experimentally compare the DP models with existing MIP and CP models, showing that, despite its nascent nature, CAASDy outperforms MIP and CP on a number of common problem classes.

Grouped variable selection with discrete optimization: Computational and statistical perspectives

We present a new algorithmic framework for grouped variable selection that is based on discrete mathematical optimization. While there exist several appealing approaches based on convex relaxations and nonconvex heuristics, we focus on optimal solutions for the ℓ0-regularized formulation, a problem that is relatively unexplored due to computational challenges. Our methodology covers both high-dimensional linear regression and nonparametric sparse additive modeling with smooth components. Our algorithmic framework consists of approximate and exact algorithms. The approximate algorithms are based on coordinate descent and local search, with runtimes comparable to popular sparse learning algorithms. Our exact algorithm is based on a standalone branch-and-bound (BnB) framework, which can solve the associated mixed integer programming (MIP) problem to certified optimality. By exploiting the problem structure, our custom BnB algorithm can solve to optimality problem instances with 5×106 features and 103 observations in minutes to hours—over 1000 times larger than what is currently possible using state-of-the-art commercial MIP solvers. We also explore statistical properties of the ℓ0-based estimators. We demonstrate, theoretically and empirically, that our proposed estimators have an edge over popular group-sparse estimators in terms of statistical performance in various regimes. We provide an open source implementation of our proposed framework.

Asymmetric probabilistic minimum-cost Hamiltonian cycle problem considering arc and vertex failures

In this paper, we introduce the asymmetric probabilistic minimum-cost Hamiltonian cycle problem (APMCHCP) where arcs and vertices in the graph are possible to fail. APMCHCP is a generalization of traditional asymmetric minimum-cost Hamiltonian cycle problem, and it has applications in many emerging areas, such as post-disaster recovery, electronic circuit design, and security maintenance of wireless sensor networks. For each vertex, we define a chance-constraint to guarantee that the probability of arriving at the vertex must be greater than or equal to a given threshold. Four mixed-integer programming (MIP) formulations are proposed for modeling the problem, including two direct formulations and two recursive formulations. A combinatorial branch-and-bound (CBB) algorithm is proposed for solving the APMCHCP, where data preprocessing steps, feasibility rules, and approaches of finding upper and lower bounds are developed. In the numerical experiments, the CBB algorithm is compared with a commercial MIP solver (Gurobi 8.0) on a test-bed of two popular benchmark instance sets. The results show that the proposed CBB algorithm significantly outperforms Gurobi solver in terms of both the size of optimally solved instances and the computing time.

The Fish Feed Production Routing Problem

This paper introduces the fish feed production routing problem (FFPRP) faced by Norwegian salmon feed producers. The FFPRP is comprised of a production scheduling problem and a rich vehicle routing problem (VRP) and thus denotes a variant of the integrated production scheduling and vehicle routing problem. We present a discrete time mixed integer programming (MIP) model of the FFPRP. Specifically, the model incorporates a multi-product, multi-trip, and multi-depot setting, where orders are produced at production lines at factories, and delivered within their respective time windows to customers by a heterogeneous fleet of vessels. The main objective is to minimize total costs, including production costs, routing costs, and costs of not delivering orders. We propose a heuristic, combining decomposition and the adaptive large neighborhood search (ALNS) heuristic, to solve the FFPRP. The decomposition-based ALNS heuristic is tested on a number of test instances which are generated based on vessel, factory, and order data from two of Norway’s largest fish feed producers. For small problem instances, for which the commercial MIP solver often is able to prove optimality, the average optimality gaps of the proposed heuristic are relatively small, while for the larger problem instances, the heuristic significantly outperforms the commercial MIP solver both in terms of solution quality and run time.

A time-dependent vessel routing problem with speed optimization

We study an operational planning problem arising in the offshore oil and gas industry, in which we determine routes, as well as sailing speeds along these routes, for a set of platform supply vessels (PSVs) servicing a given set of delivery and pickup orders such that costs are minimized. The sailing costs, mainly induced by fuel consumption for the PSVs, heavily depend on the chosen sailing speeds. Furthermore, the fuel consumption and the feasible speed ranges for the PSVs are largely affected by weather conditions that may vary over time, resulting in a weather- or Time-Dependent Vessel Routing Problem with Speed Optimization (TDVRP-SO). Optional decisions include the postponement of certain orders and the chartering of spot vessels, both associated with additional costs. We present a time-discrete mixed integer programming (MIP) model for the TDVRP-SO. To overcome the challenges of solving large-scale instances of the TDVRP-SO with a commercial MIP solver, we propose an Adaptive Large Neighborhood Search (ALNS) heuristic extended with a local search and a set partitioning model. The ALNS heuristic also includes solving the sub-problem of determining the optimal sailing speeds along each PSV route. Computational tests on instances based on a real planning case from the Norwegian continental shelf show that the ALNS heuristic efficiently provides high-quality solutions. It is also demonstrated that, in contrast to current planning practice, accounting for speed optimization and weather conditions significantly improves the solutions.

Maximum weighted induced forests and trees: new formulations and a computational comparative review

AbstractGiven a graph with a weight associated with each vertex , the maximum weighted induced forest problem (MWIF) consists of encountering a maximum weighted subset of the vertices such that induces a forest. This NP‐hard problem is known to be equivalent to the minimum weighted feedback vertex set problem, which has applicability in a variety of domains. The closely related maximum weighted induced tree problem (MWIT), on the other hand, requires that the subset induces a tree. We propose two new integer programming formulations with an exponential number of constraints and branch‐and‐cut procedures for MWIF. Computational experiments using benchmark instances are performed comparing several formulations, including the newly proposed approaches and those available in the literature, when solved by a standard commercial mixed integer programming solver. More specifically, five formulations are compared, two compact ones (i.e., with a polynomial number of variables and constraints) and the three others with an exponential number of constraints. The experiments show that a new formulation for the problem based on directed cutset inequalities for eliminating cycles (DCUT) offers stronger linear relaxation bounds earlier in the search process. The results also indicate that the other new formulation, denoted tree with cycle elimination (TCYC), outperforms those available in the literature when it comes to the average times for proving optimality for the benchmark instances with up to 81 vertices. Additionally, this formulation can achieve much lower average times for solving the larger random instances that can be optimally solved. Furthermore, we show how the formulations for MWIF can be easily extended for MWIT. Such extension allowed us to analyze the difference between the optimal solution values of the two problems when considering different classes of graphs.

Sparse regression at scale: branch-and-bound rooted in first-order optimization

We consider the least squares regression problem, penalized with a combination of the \(\ell _{0}\) and squared \(\ell _{2}\) penalty functions (a.k.a. \(\ell _0 \ell _2\) regularization). Recent work shows that the resulting estimators enjoy appealing statistical properties in many high-dimensional settings. However, exact computation of these estimators remains a major challenge. Indeed, modern exact methods, based on mixed integer programming (MIP), face difficulties for problems where the number of features \(p \sim 10^4\). In this work, we present a new exact MIP framework for \(\ell _0\ell _2\)-regularized regression that can scale to \(p \sim 10^7\), achieving speedups of at least 5000x, compared to state-of-the-art exact methods. Unlike recent work, which relies on commercial MIP solvers, we design a specialized nonlinear branch-and-bound (BnB) framework, by critically exploiting the problem structure. A key distinguishing component in our framework lies in efficiently solving the node relaxations using a specialized first-order method, based on coordinate descent (CD). Our CD-based method effectively leverages information across the BnB nodes through using warm starts, active sets, and gradient screening. In addition, we design a novel method for obtaining dual bounds from primal CD solutions, which certifiably works in high dimensions. Experiments on synthetic and real high-dimensional datasets demonstrate that our framework is not only significantly faster than the state of the art, but can also deliver certifiably optimal solutions to statistically challenging instances that cannot be solved with existing methods. We open source the implementation through our toolkit L0BnB.

Multi-supply multi-capacitated p-median location optimization via a hybrid bi-level intelligent algorithm

The Capacitated P-Median Problem (CPMP) is a classic discrete facility location problem in which p capacitated facilities (medians) are selected from an economic point of view to serve a set of demand vertexes so that the total demand assigned to each of the facilities (medians) does not exceed its capacity. Since the capacity used by each facility is fixed and does not depend on the vertex demand, the traditional capacitated p-median problem’s total capacity is underutilized. To make full use of the total capacity, a more generalized model, namely, the Multi-supply Multi-Capacitated Location Problem (MMCLP), is considered in this paper, which assumes not only that each facility is allowed to open with different capacities while meeting the constraint of total capacity but also that each demand vertex can accept supplies from different facilities. To solve the MMCLP effectively, we propose a hybrid bi-level optimization framework based on decomposition. First, the original MMCLP is decomposed into numerous mixed-integer programming (MIP) sub-problems, i.e., inner optimization is performed, and for outer optimization, a Genetic Algorithm based on Co-Evolution of Population and Neighborhood (GACEPN) is proposed to find better sub-problems near the good sub-problems among all the candidate sub-problems. The inner optimization problem is solved by a commercial MIP solver. The results obtained are returned to the outer layer so that the neighborhood of the outer layer optimization can be adjusted appropriately to generate better sub-problems, thereby approaching the optimal solution of the original problem. The experiments show that for a large-scale MMCLP, compared to the Gurobi solver, the proposed bi-level algorithm (a hybrid of the GACEPN and Gurobi) can find a better solution in less time.

Mixed Integer Programming versus Evolutionary Computation for Optimizing a Hard Real-World Staff Assignment Problem

Assigning staff to engagements according to hard constraints while optimizing several objectives is a task encountered by many companies on a regular basis. Simplified versions of such assignment problems are NP-hard. Despite this, a typical approach to solving them consists of formulating them as mixed integer programming (MIP) problems and using a stateof-the-art solver to get solutions that closely approximate the optimum.In this paper, we consider a complex real-world staff assignment problem encountered by the professional service company KPMG, with the goal of finding an algorithm that solves it faster and with a better solution than a commercial MIP solver. We follow the evolutionary algorithm (EA) metaheuristic and design a search heuristic which iteratively improves a solution using domain-specific mutation operators. Furthermore, we use a flow algorithm to optimally solve a subproblem, which tremendously reduces the search space for the EA.For our real-world instance of the assignment problem, given the same total time budget of 100 hours, a parallel EA approach finds a solution that is only 1.7% away from an upper bound for the (unknown) optimum within under five hours, while the MIP solver Gurobi still has a gap of 10.5%.

Isolation branching: a branch and bound algorithm for the k-terminal cut problem

In the k-terminal cut problem, we are given a graph with edge weights and k distinct vertices called “terminals.” The goal is to remove a minimum weight collection of edges from the graph such that there is no path between any pair of terminals. The k-terminal cut problem is NP-hard. The k-terminal cut problem has been extensively studied and a number of algorithms have been devised for it. Most are approximation algorithms. There are also fixed-parameter tractable algorithms, but none have been shown empirically practical. It is also possible to apply implicit enumeration using any integer programming formulation of the problem and solve it with a branch-and-bound algorithm. Here, we present a branch-and-bound algorithm for the k-terminal cut problem which does not rely on an integer programming formulation. Our algorithm employs “minimum isolating cuts” and, for this reason, we call our branch-and-bound algorithm Isolation Branching. In an empirical experiment, we compare the performance of Isolation Branching to that of a branch-and-bound applied to the strongest known integer programming formulation of k-terminal cut. The integer programming branch-and-bound procedure is implemented with Gurobi, a commercial mixed-integer programming solver. We compare the performance of the two approaches for real-world instances and simulated data. The results on real data indicate that Isolation Branching, coded in Python, runs an order of magnitude faster than Gurobi for problems of sizes of up to tens of thousands of vertices and hundreds of thousands of edges. Our results on simulated data also indicate that Isolation Branching scales more effectively. Though we primarily focus on creating a practical tool for k-terminal cut, as a byproduct of our algorithm we prove that the complexity of Isolation Branching is fixed-parameter tractable with respect to the size of the optimal solution, thus providing an alternative, constructive, and somewhat simpler, proof of this fact.

Two-stage stochastic programming for interdisciplinary pain management

The goal of this research is to find an optimal adaptive treatment strategy to assist physicians in prescribing treatments for patients with chronic pain. This research proposes a two-stage stochastic programming (2SP) method to optimize a treatment procedure for interdisciplinary pain management. The 2SP model incorporates non-convex nonlinear mixed integer constraints, which are constructed based on data from a real pain management program. We derive a piecewise linear approximation method to approximate the non-convex nonlinear constraints in the 2SP model. Consequently, we formulate an equivalent mixed integer linear programming (MILP) model and then solve it using a commercial mixed-integer programming solver. A comparison of the policies generated by the MILP model with the policies generated by the original nonlinear 2SP model shows that, given limited CPU time, the policies generated by the MILP model outperform those of the original nonlinear 2SP model.

Minimizing flowtime in a flowshop scheduling problem with a biased random-key genetic algorithm

In this paper, we advance the state of the art for solving the Permutation Flowshop Scheduling Problem with total flowtime minimization. For this purpose, we propose a Biased Random-Key Genetic Algorithm (BRKGA) introducing on it a new feature called shaking. With the shaking, instead to full reset the population to escape from local optima, the shaking procedure perturbs all individuals from the elite set and resets the remaining population. We compare results for the standard and the shaking BRKGA with results from the Iterated Greedy Search, the Iterated Local Search, and a commercial mixed integer programming solver, in 120 traditional instances. For all algorithms, we use warm start solutions produced by the state-of-the-art Beam-Search procedure. Computational experiments show the efficiency of proposed BRKGA, in addition to identify lower and upper bounds, as well as some optimal values, among the solutions.

ParaXpress: an experimental extension of the FICO Xpress-Optimizer to solve hard MIPs on supercomputers

The Ubiquity Generator (UG) is a general framework for the external parallelization of mixed integer programming (MIP) solvers. In this paper, we present ParaXpress, a distributed memory parallelization of the powerful commercial MIP solver FICO Xpress. Besides sheer performance, an important feature of Xpress is that it provides an internal parallelization for shared memory systems. When aiming for a best possible performance of ParaXpress on a supercomputer, the question arises how to balance the internal Xpress parallelization and the external parallelization by UG against each other. We provide computational experiments to address this question and we show computational results for running ParaXpress on a Top500 supercomputer, using up to 43,344 cores in parallel.

Heuristics for a flowshop scheduling problem with stepwise job objective function

In this work, we introduce the Flowshop Scheduling Problem with Delivery Dates and Cumulative Payoffs. This problem is a variation of the flowshop scheduling problem with job release dates that maximizes the total payoff with a stepwise job objective function. This paper contributes towards proposing a mathematical formulation for this new problem and an original constructive heuristic. Additionally, we develop a new benchmark of 300 hard instances which are available in a public repository. Besides, we provide primal and dual bounds for them. Extensive computational experiments were carried out taking into account classical constructive heuristics and hybrid local searches. Results show the merit of the FF heuristic when compared to other classical heuristics for flowshop scheduling problems. Additionally, we compare an Iterated Local Search (ILS), an Iterated Greedy Search (IGS), and a Biased Random-Key Genetic Algorithm (BRKGA), both customized for the studied problem, and a commercial mixed integer programming solver. The comparison between these methods showed that the BRKGA starting with a solution from FF heuristic is able to find the best solutions in a very short period of time. Iterated local Search and the commercial solver presented significantly worse results than the aforementioned methods.

Pseudo basic steps: bound improvement guarantees from Lagrangian decomposition in convex disjunctive programming

An elementary, but fundamental, operation in disjunctive programming is a basic step, which is the intersection of two disjunctions to form a new disjunction. Basic steps bring a disjunctive set in regular form closer to its disjunctive normal form and, in turn, produce relaxations that are at least as tight. An open question is: What are guaranteed bounds on the improvement from a basic step? In this paper, using properties of a convex disjunctive program’s hull reformulation and multipliers from Lagrangian decomposition, we introduce an operation called a pseudo basic step and use it to provide provable bounds on this improvement along with techniques to exploit this information when solving a disjunctive program as a convex MINLP. Numerical examples illustrate the practical benefits of these bounds. In particular, on a set of K-means clustering instances, we make significant bound improvements relative to state-of-the-art commercial mixed-integer programming solvers.

A hybrid primal heuristic for finding feasible solutions to mixed integer programs

We present a new framework for finding feasible solutions to mixed integer programs (MIP). We use the feasibility pump heuristic coupled to a biased random-key genetic algorithm (BRKGA). The feasibility pump heuristic attempts to find a feasible solution to a MIP by first rounding a solution to the linear programming (LP) relaxation to an integer (but not necessarily feasible) solution and then projecting it to a feasible solution to the LP relaxation. The BRKGA is able to build a pool of projected and rounded but not necessarily feasible solutions and to combine them using information from previous projections. This information is also used to fix variables during the process to speed up the solution of the LP relaxations, and to reduce the problem size in enumeration phases. Experimental results show that this approach is able to find feasible solutions for instances where the original feasibility pump or a commercial mixed integer programming solver often fail.

THE AIRPORT GATE ASSIGNMENT PROBLEM – MULTI-OBJECTIVE OPTIMIZATION VERSUS EVOLUTIONARY MULTI-OBJECTIVE OPTIMIZATION

In this paper, we approach the Airport Gate Assignment Problem by Multiobjective Optimization as well as Evolutionary Multi-objective Optimization. We solve a bi-criteria formulation of this problem by the commercial mixedinteger programming solver CPLEX and a dedicated Evolutionary Multiobjective Optimization algorithm. To deal with multiple objectives, we apply a methodology that we developed earlier to capture decision-maker preferences in multi-objective environments. We present the results of numerical tests for these two approaches.

A Benders decomposition-based matheuristic for the Cardinality Constrained Shift Design Problem

The Shift Design Problem is an important optimization problem which arises when scheduling personnel in industries that require continuous operation. Based on the forecast, required staffing levels for a set of time periods, a set of shift types that best covers the demand must be determined. A shift type is a consecutive sequence of time periods that adheres to legal and union rules and can be assigned to an employee on any day. In this paper we introduce the Cardinality Constrained Shift Design Problem; a variant of the Shift Design Problem in which the number of permitted shift types is bounded by an upper limit. We present an integer programming model for this problem and show that its structure lends itself very naturally to Benders decomposition. Due to convergence issues with a conventional implementation, we propose a matheuristic based on Benders decomposition for solving the problem. Furthermore, we argue that an important step in this approach is finding dual alternative optimal solutions to the Benders subproblems and describe an approach to obtain a diverse set of these. Numerical tests show that the described methodology significantly outperforms a commercial mixed integer programming solver on instances with 1241 different shift types and remains competitive for larger cases with 2145 shift types. On all classes of problems the heuristic is able to quickly find good solutions.

A model for solving the prescribed burn planning problem.

The increasing frequency of destructive wildfires, with a consequent loss of life and property, has led to fire and land management agencies initiating extensive fuel management programs. This involves long-term planning of fuel reduction activities such as prescribed burning or mechanical clearing. In this paper, we propose a mixed integer programming (MIP) model that determines when and where fuel reduction activities should take place. The model takes into account multiple vegetation types in the landscape, their tolerance to frequency of fire events, and keeps track of the age of each vegetation class in each treatment unit. The objective is to minimise fuel load over the planning horizon. The complexity of scheduling fuel reduction activities has led to the introduction of sophisticated mathematical optimisation methods. While these approaches can provide optimum solutions, they can be computationally expensive, particularly for fuel management planning which extends across the landscape and spans long term planning horizons. This raises the question of how much better do exact modelling approaches compare to simpler heuristic approaches in their solutions. To answer this question, the proposed model is run using an exact MIP (using commercial MIP solver) and two heuristic approaches that decompose the problem into multiple single-period sub problems. The Knapsack Problem (KP), which is the first heuristic approach, solves the single period problems, using an exact MIP approach. The second heuristic approach solves the single period sub problem using a greedy heuristic approach. The three methods are compared in term of model tractability, computational time and the objective values. The model was tested using randomised data from 711 treatment units in the Barwon-Otway district of Victoria, Australia. Solutions for the exact MIP could be obtained for up to a 15-year planning only using a standard implementation of CPLEX. Both heuristic approaches can solve significantly larger problems, involving 100-year or even longer planning horizons. Furthermore there are no substantial differences in the solutions produced by the three approaches. It is concluded that for practical purposes a heuristic method is to be preferred to the exact MIP approach.