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5.3 Mixed Integer Nonlinear Programming Models

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.

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5.3 Mixed Integer Nonlinear Programming Models

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  1. 5.3 Mixed Integer Nonlinear Programming Models

  2. A Typical MINLP Model

  3. 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.

  4. 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.

  5. 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.

  6. 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.

  7. Constraints of Example Problem

  8. 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

  9. 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).

  10. NLP Subproblem

  11. MILP Subproblem

  12. The Role of New Variable in the MILP Sub-problem

  13. 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.

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