Abstract
We consider what we term existence-constrained semi-infinite programs. They contain a finite number of (upper-level) variables, a regular objective, and semi-infinite existence constraints. These constraints assert that for all (medial-level) variable values from a set of infinite cardinality, there must exist (lower-level) variable values from a second set that satisfy an inequality. Existence-constrained semi-infinite programs are a generalization of regular semi-infinite programs, possess three rather than two levels, and are found in a number of applications. Building on our previous work on the global solution of semi-infinite programs (Djelassi and Mitsos in J Glob Optim 68(2):227–253, 2017), we propose (for the first time) an algorithm for the global solution of existence-constrained semi-infinite programs absent any convexity or concavity assumptions. The algorithm is guaranteed to terminate with a globally optimal solution with guaranteed feasibility under assumptions that are similar to the ones made in the regular semi-infinite case. In particular, it is assumed that host sets are compact, defining functions are continuous, an appropriate global nonlinear programming subsolver is used, and that there exists a Slater point with respect to the semi-infinite existence constraints. A proof of finite termination is provided. Numerical results are provided for the solution of an adjustable robust design problem from the chemical engineering literature.
Highlights
Semi-infinite programs (SIPs) are mathematical programs that have finitely many variables and infinitely many constraints
Rather than being subject to infinitely many inequalities, existence-constrained semi-infinite programs (ESIPs) are subject to infinitely many existence constraints that in turn assert that an inequality can be satisfied by at least one solution from a set of infinite cardinality
We derive the subproblems of the algorithm in [27] for the ESIP case in order to restate the algorithm for the solution of ESIPs
Summary
Semi-infinite programs (SIPs) are mathematical programs that have finitely many variables and infinitely many constraints. Following a discretization approach, Tsoukalas and Rustem [26] propose a discretized oracle problem that evaluates whether or not a given target objective value can be attained by an SIP They construct an algorithm based on a binary search of the objective space that is guaranteed to solve SIPs globally with guaranteed feasibility finitely under slightly stronger assumptions than the algorithm in [24]. The proposed discretization scheme prescribes to discretize what we term the medial-level variables while associating with each discretization point a vector of lower-level variables Following this scheme, we derive the subproblems of the algorithm in [27] for the ESIP case in order to restate the algorithm for the solution of ESIPs. Convergence and finite termination of the algorithm is guaranteed under similar assumptions as in the SIP case with substantial differences arising exclusively from the fact that the revised assumptions need to take the presence of a third level into account.
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