An Improved Branch and Bound Algorithm for Mixed Integer Nonlinear Programs. 2 Heuristics for Detecting Fractional Solutions 3 Generating Lower Bounds 4 an Improved Branch and Bound Algorithm

Introduction This paper is concerned with zero{one mixed integer nonlinear programming problems of the form (MINLP) min f(x; y) subject to g(x; y) 0 x 2 f0; 1g m y u y l Here x is a vector of m zero{one variables, y is a vector of n continuous variables, and u and l are vectors of upper and lower bounds for the continuous variables y. The objective function f and the constraint functions g are assumed to be convex. Mixed integer non-linear programs of this form arise in a number of applications , in areas as diverse as network design 13, 19], chemical process synthesis 5, 9, 16], product marketing 6], and capital budgeting 17, 20, 23]. Methods for solving mixed integer nonlinear programming problems are surveyed in 10, 12]. Branch and bound algorithms such as the ones described in 11, 17, 18, 22] work by explicitly enumerating possible values of the zero{one variables until an optimal integer solution has been found. A branch and bound algorithm begins by solving the continuous relaxation of the original problem. If a zero{one variable is fractional at optimality, the algorithm constructs two new subproblems, in which the variable is xed at zero or one. The algorithm continues solving subproblems and creating more subproblems as needed until all subproblems have been eliminated from consideration. A subproblem can be eliminated from consideration if it is infeasible, if the solution to the subproblem has a higher objective value than a known integer solution, or if the solution to the subproblem is an integer solution. After all cases have been considered, the optimal solution is simply the best integer solution that was discovered while solving the subprob-lems. This paper describes an improved branch and bound algorithm that uses heuristics to determine when to split a problem into subproblems and that uses a lower bounding procedure to eliminate subproblems from consideration. The remainder of this paper is organized as follows: In section 1, we review the sequential quadratic programming method. In section 2, we describe the heuristics that our branch and bound algorithm uses. In section 3, we describe a method for calculating a lower bound on the value of a subproblem without solving the subproblem to optimality. In section 4, we describe our branch and bound algorithm in detail. Section 5 contains computational results for a number of sample problems. Our conclusions are presented in section …