UBC Theses and Dissertations
Signomial programs with equality constraints : numerical solution and applications Yan, James
Many design problems from diverse engineering disciplines can be formulated as signomial programs with both equality and inequality constraints. However, existing computational methods of signomial programming are applicable to programs with inequality constraints only. In this thesis an algorithm is proposed and implemented to solve signomial programs with mixed inequality and equality constraints. The algorithm requires no manipulation of the constraints by the user and is compatible with the results of signomial programming. The proposed algorithm is a synthesis shaped by three concepts: the method of multipliers viewed as a primal-dual method, partial dualization, and the retention of the structure of a signomial program. The strategy of the algorithm is to replace the original problem with a sequence of subproblems each subject to the original signomial inequality constraints only. The subproblem's objective function is the original cost augmented with the equality constraints and specified by a parameter vector λ and a penalty constant K. The algorithm then alternates between solving the subproblem and updating λ and K. The convergence of the algorithm is, under suitable assumptions, guaranteed by the convergence of the method of multipliers and the method used to solve the subproblem. Because the subproblem has the form of a regular signomial program, it can in principle be solved by the techniques of signomial programming. A new numerical method implementing a variant of the Avriel-Williams algorithm for signomial programs is suggested for solving the subproblem. The method relies on monomial condensation, and the combined use of the reduced gradient method and a cutting plane scheme. Details of the method's implementation are considered, and some computational experience has been acquired. The proposed method also has the advantageous flexibility of being able to handle non-signomial differentiable objective functions. Four updating schemes for λ and K are formulated and evaluated in a series of numerical experiments. In terms of the rate of convergence, the most promising scheme tested is the use of the Hestenes-Powell rule for updating λ and the moderate monotonic increase of K after the completion of each subproblem. Convergence can also be considerably accelerated by properly scaling the equality constraints and performing only inexact minimization in the first few subproblems. The applicability of the algorithms developed in this thesis is illustrated with the solution of three design examples drawn from structural design and chemical process engineering.
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