269185 A Parametric Approach for the Global Optimization of Mixed-Integer Fractional Programs
Mixed-integer fractional programming (MIFP) problems frequently occur in process engineering, e.g. cyclic scheduling , and integration scheduling and control [2, 3]. In this work, we specifically consider the MIFPs with a convex quadratic function in the numerator and a linear function in the denominator. This is a non-convex mixed integer nonlinear programming (MINLP) problem and we developed an efficient approach for its global optimization.
First, the relaxed MIFP problem is proved to be quasi-convex. Based on the property, the MINLP solver SBB  can guarantee the global optimality. Because each nonlinear programming (NLP) subproblem in the branch-and-bound is solved to the global optimum, the global solution to the MINLP problem can be obtained by a standard branch-and-bound algorithm. Another popular MINLP solver DICOPT  uses the outer-approximation method and it might cut off some feasible solution space during the iterations. Thus, DICOPT cannot guarantee the global optimality of the solution. The global optimal solution can be also obtained by the global MINLP solvers, such as BARON  and GloMIQO .
Second, the MIFP is transformed into an equivalent parametric mixed-integer programming (PMIP) problem which is the numerator minus the denominator multiplied by a parameter. The solution to the PMIP problem is a function of the parameter, called the optimal value function. A feasible solution to the MIFP problem is the global optimal solution if and only if the solution is also the global optimal solution to the PMIP problem with the parameter which is the root of the optimal value function [8-10]. The condition for the equivalence is very general, i.e. the denominator should be positive. There are no assumptions required on the types or the convexities of the functions in the numerator and the denominators. However, the equivalence is true only for the global minima of the two problems. For the problem we consider, the equivalent PMIP problem is a convex mixed-integer quadratic programming (MIQCQ) problem and it can be solved to the global minimum by state-of-the-art solvers, such as CPLEX, in an efficient way. To guarantee the equivalence, the PMIP problem must be solved at the parameter which is the root of the optimal value function.
Third, two approaches are applied to find the root of the optimal value function which has no explicit expression. They are the bisection method  and the generalized Newton’s method . The bisection method only uses the sign information of F(q) in the iteration while the generalized Newton’s method requires the information of the subgradient. By using more information of the function, the generalized Newton’s method has a faster convergence rate than the bisection method. The Newton's method converges supper-linearly while the bisection algorithm converges linearly.
The PMIP solution approaches based on the bisection method and the Newton’s method are demonstrated by a large-scale MIFP problem derived from an integrated scheduling and control problem . The results are compared with those returned by the MINLP solvers, SBB, DICOPT, and BARON 9.3.1. The results showed that the proposed approaches can globally optimize the MIFPs within much shorter times than general-purpose MINLP methods.
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