Global solution of semi-infinite programs

  • Authors:

  • B. Bhattacharjee;P. Lemonidis;W. H. Green, Jr.;P. I. Barton

  • Affiliations:

  • Dept. of Chemical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, USA;Dept. of Chemical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, USA;Dept. of Chemical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, USA;Dept. of Chemical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, USA

  • Venue:
  • Mathematical Programming: Series A and B
  • Year:
  • 2005

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Abstract

Optimization problems involving a finite number of decision variables and an infinite number of constraints are referred to as semi-infinite programs (SIPs). Existing numerical methods for solving nonlinear SIPs make strong assumptions on the properties of the feasible set, e.g., convexity and/or regularity, or solve a discretized approximation which only guarantees a lower bound to the true solution value of the SIP. Here, a general, deterministic algorithm for solving semi-infinite programs to guaranteed global optimality is presented. A branch-and-bound (B&B) framework is used to generate convergent sequences of upper and lower bounds on the SIP solution value. The upper-bounding problem is generated by replacing the infinite number of real-valued constraints with a finite number of constrained inclusion bounds; the lower-bounding problem is formulated as a convex relaxation of a discretized approximation to the SIP. The SIP B&B algorithm is shown to converge finitely to ɛ−optimality when the subdivision and discretization procedures used to formulate the node subproblems are known to retain certain convergence characteristics. Other than the properties assumed by globally-convergent B&B methods (for finitely-constrained NLPs), this SIP algorithm makes only one additional assumption: For every minimizer x* of the SIP there exists a sequence of Slater points xn for which ** (cf. Section 5.4). Numerical results for test problems in the SIP literature are presented. The exclusion test and a modified upper-bounding problem are also investigated as heuristic approaches for alleviating the computational cost of solving a nonlinear SIP to guaranteed global optimality.