Experiments With a Feasibility Pump Approach for Nonconvex MINLPs

Mathematical Programming, 2009

We present an algorithm for finding a feasible solution to a convex mixed integer nonlinear program. This algorithm, called Feasibility Pump, alternates between solving nonlinear programs and mixed integer linear programs. We also discuss how the algorithm can be iterated so as to improve the first solution it finds, as well as its integration within an outer approximation scheme. We report computational results.

2010

2020

The feasibility pump is a recent, highly successful heuristic for general mixed integer linear programming problems. We show that the feasibility pump heuristic can be interpreted as a discrete version of the proximal point algorithm. In doing so, we extend and generalize some of the fundamental results in this area to provide new supporting theory. We show that feasibility pump algorithms implicitly minimizes a weighted combination of the objective and a term which penalizes lack of integrality. This function has many local minima, some of which correspond to feasible integral solutions; the feasibility pump's use of random restarts can be viewed as seeking to escape these local minima when they are not feasible integral solutions. This interpretation suggests alternative ways of incorporating restarts, one of which is the application of cutting planes. Numerical experiments with cutting planes show encouraging results on standard test libraries.

Optimization Methods and Software, 2019

The solver DICOPT is based on the outer-approximation algorithm used for solving mixed-integer nonlinear programming (MINLP) problems. This algorithm is very effective for solving some types of convex MINLPs. However, it has been observed that DICOPT has difficulties solving instances in which some of the nonlinear constraints are so restrictive that nonlinear subproblems generated by the algorithm are infeasible. This problem is addressed in this paper with a feasibility pump algorithm, which modifies the objective function in order to efficiently find feasible solutions. It has been implemented as a preprocessing algorithm, which is used to initialize both the incumbent and the mixed-integer linear relaxation of the outer-approximation algorithm. Computational comparisons with previous versions of DICOPT on a set of convex MINLPs demonstrate the effectiveness of the proposed algorithm in terms of solution quality and solution time.

2010

Mixed-Integer optimization represents a powerful tool for modelling many optimization problems arising from real-world applications. The Feasibility pump is a heuristic for finding feasible solutions to mixedinteger linear problems. In this work, we propose a new feasibility pump approach for MIP problems using concave non differentiable penalty functions for measuring solution integrality.

2017

Feasibility pump is one of the successful heuristic solution approaches developed almost a decade ago for computing high-quality feasible solutions of single-objective integer linear programs, and it is implemented in exact commercial solvers such as CPLEX and Gurobi. In this study, we present the first Feasibility Pump Based Heuristic (FPBH) approach for approximately generating nondominated frontiers of multi-objective mixed integer linear programs with an arbitrary number of objective functions. The proposed algorithm extends our recent study for bi-objective pure integer programs that employs a customized version of several existing algorithms in the literature of both single-objective and multi-objective optimization. The method has two desirable characteristics: (1) There is no parameter to be tuned by users other than the time limit; (2) It can naturally exploit parallelism. An extensive computational study shows the efficacy of the proposed method on some existing standard t...

Surveys in Operations Research and Management Science, 2012

A wide range of problems arising in practical applications can be formulated as Mixed-Integer Nonlinear Programs (MINLPs). For the case in which the objective and constraint functions are convex, some quite effective exact and heuristic algorithms are available. When nonconvexities are present, however, things become much more difficult, since then even the continuous relaxation is a global optimisation problem. We survey the literature on non-convex MINLP, discussing applications, algorithms and software. Special attention is paid to the case in which the objective and constraint functions are quadratic.

Discrete Optimization, 2007

Finding a feasible solution of a given Mixed-Integer Programming (MIP) model is a very important (NP-complete) problem that can be extremely hard in practice. Very recently, Fischetti, Glover and Lodi proposed a heuristic scheme for finding a feasible solution to general MIPs, called Feasibility Pump (FP). According to the computational analysis reported by these authors, FP is indeed quite effective in finding feasible solutions of hard 0-1 MIPs. However, MIPs with generalinteger variables seem much more difficult to solve by using the FP approach, possibly because of the simple rounding nature of the basic method.

Computers & Operations Research, 2019

Feasibility pump is one of the successful heuristics developed almost a decade ago for computing highquality feasible solutions of single-objective integer linear programs, and it is implemented in exact commercial solvers such as CPLEX and Gurobi. In this study, we present the first Feasibility Pump Based Heuristic (FPBH) for approximately generating nondominated frontiers of multi-objective mixed integer linear programs with an arbitrary number of objectives. The proposed algorithm extends our recent study for bi-objective pure integer programs and employs a customized version of several existing algorithms in the literature of both single-objective and multi-objective optimization. The method has two desirable characteristics: (1) There is no parameter to be tuned by users other than the time limit; (2) It can naturally exploit parallelism. An extensive computational study shows the efficacy of the proposed method on some existing standard test instances in which the true frontier is known, and also some randomly generated instances. We also numerically show the importance of parallelization feature of FPBH and illustrate that FPBH outperforms MDLS developed by Tricoire (2012) on instances of multi-objective knapsack problem. We test the effect of using different commercial and non-commercial linear programming solvers for solving linear programs arising during the course of FPBH, and show that the performance of FPBH is almost the same in all cases. It is worth mentioning that FPBH is available as an open source Julia package, named as 'FPBH.jl', on GitHub. The package is compatible with the popular JuMP modeling language and supports input in LP and MPS file formats. The package can plot nondominated frontiers, can compute different quality measures (hypervolume, cardinality, coverage and uniformity), supports execution on multiple processors, and can use any linear programming solver supported by MathProgBase.jl (such as CPLEX, Clp, GLPK, etc).

Mathematical Programming Computation, 2014

The Feasibility Pump (FP) has proved to be an effective method for finding feasible solutions to mixed integer programming problems. FP iterates between a rounding procedure and a projection procedure, which together provide a sequence of points alternating between LP feasible but fractional solutions, and integer but LP relaxed infeasible solutions. The process attempts to minimise the distance between consecutive iterates, producing an integer feasible solution when closing the distance between them. We investigate the benefits of enhancing the rounding procedure with a clever integer line search that efficiently explores a large set of integer points. An extensive computational study on benchmark instances demonstrates the efficacy of the proposed approach.