5.3 Mixed Integer Nonlinear Programming Models

A Typical MINLP Model

Remarks The y’s are typically chosen to control the continuous variables x by either forcing one (or more) variable to be zero or by allowing them to assume positive values. The choice of y should be done in such a way that y appears linearly, because then the problem is much easier to solve. The set X is specified by bounds and other inequalities involving x only, whereas Y is defined by conditions that the components of y be binary or integer, plus other inequalities or equations involving y only.

Branch-and-Bound Method The same BB method used to solve MILP can be used to solve MINLP. The only difference is that for MINLP problems the relaxed subproblems at the nodes of the BB tree are continuous variable NLPs and must be solved by NLP methods. BB methods are guaranteed to solve linear or nonlinear problems if allowed to continue until the gap between upper and lower bounds reaches zero, provided that a global optimum is found for each relaxed subproblem at each node of the BB tree.

Sufficient Conditions of Convexity of Each Relaxed Subproblem The objective function f(x) is convex. Each component of h(x) is linear. Each component of g(x) is convex over the set X. The set X is convex. The set Y is determined by linear constraints and the integer restrictions on y.

Example: Optimal Selection of Processes This problem involves the manufacture of a chemical C in process 1 that uses raw material B. B can either be purchased or produced via processes 2 or 3, both of which use chemical A as a raw material. We want to determine which processes to use and their production levels in order to maximize profit.

Constraints of Example Problem

Objective Function of Example Problem Income from product sales: 13C1 Expense for the purchase of B: 7BP Expense for the purchase of A: 1.8(A2+A3) Annualized investment for the 3 processes: (3.5Y1+2C1)+(Y2+B2)+(1.5Y3+1.2B3) The objective function is profit (PR) to be maximized: PR=11C1-3.5Y1-Y2-1.5Y3-B2-1.2B3-7BP-1.8A2-1.8A3

Solving MINLP Using Outer Approximation (OA) Each major iteration of OA involves solving 2 subproblems: a continuous variable nonlinear program (NLP), and A mixed-integer linear program (MILP).

NLP Subproblem

MILP Subproblem

The Role of New Variable in the MILP Sub-problem

OA Algorithm Duran and Grossman (1986) showed that if the convexity assumptions hold, then the optimal value of MILP subproblem is an LOWER BOUND on the optimal MINLP objective value. Because a new set of linear constraints is added at each iteration, this lower bound increases (or remains the same) at each iteration. Under the convexity assumptions, the upper and lower bounds converge to the true optimal MINLP value in a finite number of iterations.