Retrospective Approximation Algorithms for Multi-Objective Simulation Optimization on Integer Lattices

Kyle Cooper, Purdue University

Abstract

We consider multi-objective simulation optimization (MOSO) problems, that is, nonlinear optimization problems in which multiple simultaneous objective functions can only be observed with stochastic error, e.g., as output from a Monte Carlo sim- ulation model. In this context, the solution to a MOSO problem is the efficient set, which is the set of all feasible decision points for which no other feasible decision point is at least as good on all objectives and strictly better on at least one objective. We are concerned primarily with MOSO problems on integer lattices, that is, MOSO problems where the feasible set is a subset of an integer lattice. In the first study, we propose the Retrospective Partitioned Epsilon-constr aint with Relaxed Local Enumeration (R-PzRLE) algorithm to solve the bi-objective sim- ulation optimization problem on integer lattices. R-PsRLE is designed for sampling efficiency. It uses a retrospective approximation (RA) framework to repeatedly call the PeRLE sample-path solver at a sequence of increasing sample sizes, using the solution from the previous RA iteration as a warm start for the current RA iteration. The PeRLE sample-path solver is designed to solve the sample-path problem only to within a tolerance commensurate with the sampling error. It comprises a call to each of the Pe and RLE algorithms, in sequence. First, Pe searches for new points to add to the sample-path local efficient set by solving multiple constrained single- objective optimization problems. Pz places constraints to locate new sample-path local efficient points that are a function of the standard error away, in the objec- tive space, from those already obtained. Then, the set of sample-path local efficient points found by Pz is sent to RLE, which is a local crawling algorithm that ensures the set is a sample-path approximate local efficient set. As the number of RA iterations increases. R-PeRLE provably converges to a local efficient set with probability one un- der appropriate regularity conditions. We also propose a naive. provably-convergent benchmark algorithm for problems with two or more objectives. called R-MINRLE. R-MINRLE is identical to R-P©RLE except that it replaces the Pe algorithm with an algorithm that updates one local minimum on each objective before invoking RLE. R-PeRLE performs favorably relative to R-MINRLE and the current state of the art. MO-COMPASS. in our numerical experiments. Our work points to a family of RA algorithms for MOSO on integer lattices that employ RLE for certification of a sample-path approximate local efficient set. and for which the convergence guarantees are provided in this study. In the second study, we present the PyMOSO software package for solving multi- objective simulation optimization problems on integer lattices, and for implementing and testing new simulation optimization (SO) algorithms. First, for solving MOSO problems on integer lattices, Py MOSO implements R-PeRLE and R-MiInRLE, which are developed in the first study. Both algorithms employ psendo-gradients, are de- signed for sampling efficiency, and return solutions that, under appropriate regularity conditions, provably converge to a local efficient set with probability one as the simu- lation budget increases. Py MOSO can interface with existing simulation software and can obtain simulation replications in parallel. Second, for implementing and testing new SO algorithms, PyMOSO includes pseudo-random number stream management, implements algorithm testing with independent pseudo-random number streams run in parallel, and computes the performance of algorithms with user-defined metrics. For convenience, we also include an implementation of R-SPLINE for problems with one objective. The PyMOSO source code is available under a permissive open source license.

Degree

Ph.D.

Advisors

Hunter, Purdue University.

Subject Area

Design|Computer science|Health care management|Management|Medicine|Operations research

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