State-of-the-Art Survey—Constrained Nonlinear 0–1 Programming

Abstract

We consider nonlinear programs in 0–1 variables with nonlinear constraints and survey the main approaches to their solution: (i) linearization; (ii) algebraic methods; (iii) enumerative methods and (iv) cutting-plane methods. We also present an extensive computational comparison of algorithms of all four categories. Enumerative methods appear to be the most promising. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

Cluster analysis and mathematical programming

“…An iterative solution scheme [37] reduces this problem to a sequence of quadratic programs in 0-1 variables. These last problems, as well as other quadratic 0-1 programs discussed above, can be solved by an algebraïc (or variable elimination) method [24], linearisation [113], cutting planes [3] or branch-and-bound [63], possibly exploiting the persistency properties of roof duality theory [56]. Combining column generation with an interior point method [41] allows solution of minimum sum-ofsquares partitioning problem with N ≤ 150.…”

The unconstrained binary quadratic programming problem: a survey

“…For general constraints, however, appropriate penalty functions are not known in advance and need to be "discovered." A simple procedure (see for instance Hammer & Rudeanu (1968); Hansen (1979);Hansen, Jaumard, & Mathon (1993);and Boros & Hammer( 2002) for finding an appropriate penalty for any linear constraint is given as follows: This model accommodates both quadratic and linear objective functions since the linear case results when Q is a diagonal matrix (observing that x j 2 = x j when x j is a 0-1 variable). Under the assumption that A and b have integer components, problems with inequality constraints can also be put in this form by representing their bounded slack variables by a binary expansion.…”

A new modeling and solution approach for the number partitioning problem

“…A suitable choice of the scalar penalty P, for the general application of this reformulation approach, can always be chosen so that the optimal solution to UQP(PEN) is the optimal solution to the original constrained problem. (Hammer and Rudeanu [7], Hansen [8], Hansen et al [9], and Boros and Hammer [2]). We illustrate the procedure in the following example.…”