The feasibility pump

Abstract. This paper provides a survey of recent progress and software for solving mixed integer nonlinear programs (MINLP) wherein the objective and constraints are defined by convex functions and integrality restrictions are imposed on a subset of the decision variables. Convex MINLPs have received sustained attention in very years. By exploiting analogies to the case of well-known techniques for solving mixed integer linear programs and incorporating these techniques into the software, significant improvements have been made in our ability to solve the problems.Key words. Mixed Integer Nonlinear Programming; Branch and Bound; AMS(MOS) subject classifications.1. Introduction. Mixed-Integer Nonlinear Programs (MINLP) are optimization problems where some of the variables are constrained to take integer values and the objective function and feasible region of the problem are described by nonlinear functions. Such optimization problems arise in many real world applications. Integer variables are often required to model logical relationships, fixed charges, piecewise linear functions, disjunctive constraints and the non-divisibility of resources. Nonlinear functions are required to accurately reflect physical properties, covariance, and economies of scale.In all its generality, MINLP forms a particularly broad class of challenging optimization problems as it combines the difficulty of optimizing over integer variables with handling of nonlinear functions. Even if we restrict our model to contain only linear functions, MINLP reduces to a Mixed-Integer Linear Program (MILP), which is an NP-Hard problem [56]. On the other hand, if we restrict our model to have no integer variable but allow for general nonlinear functions in the objective or the constraints, then MINLP reduces to a Nonlinear Program (NLP) which is also known to be NP-Hard [91]. Combining both integrality and nonlinearity can lead to examples of MINLP that are undecidable [68].

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“…It was initially proposed by Fischetti, Glover and Lodi [49] for MILP, and can be extended to convex MINLP in several manners.…”

Section: Feasibility Pump(s)mentioning

confidence: 99%

“…The procedure stops whenever an integer feasible solution is found (orx k = x k ). This basic procedure may cycle or stall without finding a integer feasible solution and randomization has been suggested to restart the procedure [49]. Several variants of this basic procedure have been proposed in the context of MILP [19,4,51].…”

Section: Feasibility Pump(s)mentioning

confidence: 99%

Algorithms and Software for Convex Mixed Integer Nonlinear Programs

Bonami

Kılınç

Linderoth

2011

Mixed Integer Nonlinear Programming

191

181

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“…The feasibility pump heuristic (Fischetti et al, 2005) can be used to find a feasible integer solution more quickly, which may be helpful to solve the MILP transshipment model faster. Two types of feasibility pump heuristics are implemented in CPLEX: a feasibility pump with an emphasis on finding a feasible solution (fpheur 1), and one with an emphasis on finding a feasible solution with a good objective value (fpheur 2) (CPLEX, 2013).…”

Section: Feasibility Pumpmentioning

confidence: 99%

Computational strategies for large-scale MILP transshipment models for heat exchanger network synthesis

Yang

Grossmann

Miller

2015

Computers & Chemical Engineering

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Determining the minimum number of units is an important step in heat exchanger network synthesis (HENS). The MILP transshipment model (Papoulias and Grossmann, 1983) and transportation model (Cerda and Westerberg, 1983b) were developed for this purpose. However, they are computationally expensive when solving for large-scale problems. Several approaches are studied in this paper to enable the fast solution of large-scale MILP transshipment models. Model reformulation techniques are developed for tighter formulations with reduced LP relaxation gaps. Solution strategies are also proposed for improving the efficiency of the branch and bound method. Both approaches aim at finding the exact global optimal solution with reduced solution times. Several approximation approaches are also developed for finding good approximate solutions in relatively short times. Case study results show that the MILP transshipment model can be solved for relatively large-scale problems in reasonable times by applying the approaches proposed in this paper. • Several approximation approaches are developed for finding near optimal solutions. Key Words• Some relatively large-scale MILP transshipment models are solved in reasonable times by using the approaches proposed in this study.

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“…The fix-and-relax heuristic applied to the CTA problem other approaches for initial good solutions in MILPs, such as the feasibility pump [17], but as it will seen in Section 4, in practice FR outperformed them. FR can be briefly stated as follows.…”

Section: Fix-and-relaxmentioning

confidence: 99%

“…FP [17] is considered an efficient heuristic for the fast computation of hopefully good initial feasible solutions to MILPs. We used the objective feasibility pump (oFP) [1], which is more efficient than FP in terms of quality of the solution and the analytic center feasibility pump (AC-FP) [2], which was introduced as a good alternative in some MILP instances (either in time or quality of the solution).…”

Section: Fix-and-relax and Feasibility Pump Heuristicsmentioning

confidence: 99%

Fix-and-relax approaches for controlled tabular adjustment

Baena

Castro

González

2015

Computers & Operations Research

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Controlled tabular adjustment (CTA) is a relatively new protection technique for tabular data protection. CTA formulates a mixed integer linear programming problem, which is challenging for tables of moderate size. Even finding a feasible initial solution may be a challenging task for large instances. On the other hand, end users of tabular data protection techniques give priority to fast executions and are thus satisfied in practice with suboptimal solutions. This work has two goals. First, the fix-and-relax (FR) strategy is applied to obtain good feasible initial solutions to large CTA instances. FR is based on partitioning the set of binary variables into clusters to selectively explore a smaller branch-and-cut tree. Secondly, the FR solution is used as a warm start for a block coordinate descent (BCD) heuristic (approach named FR+BCD); BCD was confirmed to be a good option for large CTA instances in an earlier paper by the second and third co-authors (Computers & Operations Research 2011). We report extensive computational results on a set of real-world and synthetic CTA instances. FR is shown to be competitive compared to CPLEX branch-and-cut in terms of quickly finding either a feasible solution or a good upper bound. FR+BCD improved the quality of FR solutions for approximately 25% and 50% of the synthetic and real-world instances, respectively. FR or FR+BCD provided similar or better solutions in less CPU time than CPLEX for 73% of the difficult real-world instances.

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The feasibility pump

Cited by 308 publications

References 12 publications

Algorithms and Software for Convex Mixed Integer Nonlinear Programs

Algorithms and Software for Convex Mixed Integer Nonlinear Programs

Computational strategies for large-scale MILP transshipment models for heat exchanger network synthesis

Fix-and-relax approaches for controlled tabular adjustment

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