This chapter introduces the notation used in the book and discusses the mixed integer programming (MIP) computational framework in which heuristics are developed, used, and evaluated. The chapter starts by formally definining MIP and presenting the basic complete algorithms to solve it. Then, the more important building block concepts at the core of primal heuristics are presented, as well as the way in which they are incorporated in the MIP framework and their impact.

This chapter reviews the a large family of relatively cheap primal heuristics that generally try to convert infeasible solutions obtained by solving the continuous relaxation of a MIP into feasible solutions. The review is conducted by following three main concepts, namely that of rounding a fractional point to an integer one, that of propagating the logical implication of a decision on a variable to other variables, and that of diving, i.e., sequentially make decisions on variables. The combinations of these concepts are extensively analyzed.

The computational study presented in this chapter analyzes the impact of primal heuristics from different angles. This is done by investigating in which respect primal heuristics have an impact on the performance of a MIP solver, with respect to multiple performance measures.

This chapter presents the primal heuristics in the feasibility pump family. The fundamental idea of all feasibility pump algorithms is to construct two sequences of points that hopefully converge to a feasible solution of a given optimization problem. The points in the first sequence are feasible with respect to the linear programming constraints of the MIP, while those in the second sequence respect the integrality requirements. This basic concept has been developed in many ways in the literature, and this chapter gives an exhaustive overview of the resulting algorithms.

This chapter concerns the vast family of large neighborhood search primal heuristics. These are local search heuristics that generally assume the knowledge of one or more feasible MIP solutions and explore "large" neighborhoods in the attempt to improve the incumbent, i.e., the best feasible solution computed so far by the MIP algorithm. A neighborhood is large if, in general, it cannot be explored by complete enumeration, so the various techniques developed for defining those neighborhoods and exploring them are discussed.

This chapter discusses the extension of many primal heuristics developed for MIP to mixed integer nonlinear programming, a larger and even more challenging class of mathematical optimization problems that contains MIP. The importance of primal heuristics for this area is highlighted and some novel ideas originated from specifically considering mixed integer nonlinear programs are also reviewed.

This chapter discusses some primal heuristics that do not necessarily belong with the mainstream methods that have been implemented in the MIP solvers but are interesting either for historical reasons (first attempts of the MIP community to devise heuristic solutions within a general MIP scheme) or because they combine many of the ingredients that are at the core of this book.

The integration of machine learning models within MIP computation has been an exciting research trend in the last decade. This chapter reviews the use of such models in conjunction with primal heuristics for MIP.

This chapter covers the basics from real analysis to linear algebra and the theory of computation that is foundational for the rest of the book. A careful discussion of different models of computation is taken up, which discusses several issues that are often ignored in other presentations of optimization theory and algorithms.

A careful exposition of the conceptual underpinnings of algorithmic or computational optimization is presented. Computation in continuous optimization has its origins in the traditions of scientific computing and numerical analysis, whereas discrete optimization broadly views computation via the Turing machine model. The different views lead to some friction. In the continuous world, one often designs algorithms assuming one can perform exact operations with real numbers (consider, for example, Newton’s method), which is impossible in the Turing machine model. In the discrete world, the “input" to a Turing machine becomes a tricky question when dealing with general nonlinear functions and sets. The question of “complexity" of an optimization algorithm is also treated in somewhat different ways in the two communities. This chapter, combined with the careful discussion of computation models in Chapter 1, shows how all these issues can be handled in a unified, coherent way making no distinction whatsoever between "continuous" and "discrete" optimization.