A stochastic approximation method for approximating the efficient frontier of chance-constrained nonlinear programs
We propose a stochastic approximation method for approximating the efficient frontier of chance-constrained nonlinear programs. Our approach is based on a bi-objective viewpoint of chance-constrained programs that seeks solutions on the efficient frontier of optimal objective value versus risk of co...
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| Published in | Mathematical programming computation Vol. 13; no. 4; pp. 705 - 751 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
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Springer Berlin Heidelberg
01.12.2021
Springer Nature B.V |
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| Abstract | We propose a stochastic approximation method for approximating the efficient frontier of chance-constrained nonlinear programs. Our approach is based on a bi-objective viewpoint of chance-constrained programs that seeks solutions on the efficient frontier of optimal objective value versus risk of constraints violation. To this end, we construct a reformulated problem whose objective is to minimize the probability of constraints violation subject to deterministic convex constraints (which includes a bound on the objective function value). We adapt existing smoothing-based approaches for chance-constrained problems to derive a convergent sequence of smooth approximations of our reformulated problem, and apply a projected stochastic subgradient algorithm to solve it. In contrast with exterior sampling-based approaches (such as sample average approximation) that approximate the original chance-constrained program with one having finite support, our proposal converges to stationary solutions of a smooth approximation of the original problem, thereby avoiding poor local solutions that may be an artefact of a fixed sample. Our proposal also includes a tailored implementation of the smoothing-based approach that chooses key algorithmic parameters based on problem data. Computational results on four test problems from the literature indicate that our proposed approach can efficiently determine good approximations of the efficient frontier. |
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| AbstractList | We propose a stochastic approximation method for approximating the efficient frontier of chance-constrained nonlinear programs. Our approach is based on a bi-objective viewpoint of chance-constrained programs that seeks solutions on the efficient frontier of optimal objective value versus risk of constraints violation. To this end, we construct a reformulated problem whose objective is to minimize the probability of constraints violation subject to deterministic convex constraints (which includes a bound on the objective function value). We adapt existing smoothing-based approaches for chance-constrained problems to derive a convergent sequence of smooth approximations of our reformulated problem, and apply a projected stochastic subgradient algorithm to solve it. In contrast with exterior sampling-based approaches (such as sample average approximation) that approximate the original chance-constrained program with one having finite support, our proposal converges to stationary solutions of a smooth approximation of the original problem, thereby avoiding poor local solutions that may be an artefact of a fixed sample. Our proposal also includes a tailored implementation of the smoothing-based approach that chooses key algorithmic parameters based on problem data. Computational results on four test problems from the literature indicate that our proposed approach can efficiently determine good approximations of the efficient frontier. |
| Author | Luedtke, James R. Kannan, Rohit |
| Author_xml | – sequence: 1 givenname: Rohit surname: Kannan fullname: Kannan, Rohit email: rohitk@alum.mit.edu organization: Wisconsin Institute for Discovery, University of Wisconsin-Madison – sequence: 2 givenname: James R. surname: Luedtke fullname: Luedtke, James R. organization: Department of Industrial and Systems Engineering and Wisconsin Institute for Discovery, University of Wisconsin-Madison |
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| Keywords | Efficient frontier Stochastic approximation Chance constraints 90C29 90C15 Stochastic subgradient |
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| Title | A stochastic approximation method for approximating the efficient frontier of chance-constrained nonlinear programs |
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