Stochastic dual dynamic programming for multistage stochastic mixed-integer nonlinear optimization

In this paper, we study multistage stochastic mixed-integer nonlinear programs (MS-MINLP). This general class of problems encompasses, as important special cases, multistage stochastic convex optimization with non-Lipschitzian value functions and multistage stochastic mixed-integer linear optimizati...

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Vydané v:	Mathematical programming Ročník 196; číslo 1-2; s. 935 - 985
Hlavní autori:	Zhang, Shixuan, Sun, Xu Andy
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2022 Springer Springer Nature B.V
Predmet:	Algorithms Calculus of Variations and Optimal Control; Optimization Combinatorics Complexity Computational geometry Convexity Dynamic programming Full Length Paper Global optimization Investment analysis Mathematical and Computational Physics Mathematical Methods in Physics Mathematical optimization Mathematics Mathematics and Statistics Mathematics of Computing Mixed integer Numerical Analysis Optimization Regularization Sampling Theoretical Complexity analysis 90C30 90C11 90C39 MINLP 90C15 49N15 Multistage stochastic programming SDDP 90C60
ISSN:	0025-5610, 1436-4646
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Abstract	In this paper, we study multistage stochastic mixed-integer nonlinear programs (MS-MINLP). This general class of problems encompasses, as important special cases, multistage stochastic convex optimization with non-Lipschitzian value functions and multistage stochastic mixed-integer linear optimization. We develop stochastic dual dynamic programming (SDDP) type algorithms with nested decomposition, deterministic sampling, and stochastic sampling. The key ingredient is a new type of cuts based on generalized conjugacy. Several interesting classes of MS-MINLP are identified, where the new algorithms are guaranteed to obtain the global optimum without the assumption of complete recourse. This significantly generalizes the classic SDDP algorithms. We also characterize the iteration complexity of the proposed algorithms. In particular, for a ( T + 1 ) -stage stochastic MINLP satisfying L -exact Lipschitz regularization with d -dimensional state spaces, to obtain an ε -optimal root node solution, we prove that the number of iterations of the proposed deterministic sampling algorithm is upper bounded by O ( ( 2 L T ε ) d ) , and is lower bounded by O ( ( LT 4 ε ) d ) for the general case or by O ( ( LT 8 ε ) d / 2 - 1 ) for the convex case. This shows that the obtained complexity bounds are rather sharp. It also reveals that the iteration complexity depends polynomially on the number of stages. We further show that the iteration complexity depends linearly on T , if all the state spaces are finite sets, or if we seek a ( T ε ) -optimal solution when the state spaces are infinite sets, i.e. allowing the optimality gap to scale with T . To the best of our knowledge, this is the first work that reports global optimization algorithms as well as iteration complexity results for solving such a large class of multistage stochastic programs. The iteration complexity study resolves a conjecture by the late Prof. Shabbir Ahmed in the general setting of multistage stochastic mixed-integer optimization.
AbstractList	In this paper, we study multistage stochastic mixed-integer nonlinear programs (MS-MINLP). This general class of problems encompasses, as important special cases, multistage stochastic convex optimization with non-Lipschitzian value functions and multistage stochastic mixed-integer linear optimization. We develop stochastic dual dynamic programming (SDDP) type algorithms with nested decomposition, deterministic sampling, and stochastic sampling. The key ingredient is a new type of cuts based on generalized conjugacy. Several interesting classes of MS-MINLP are identified, where the new algorithms are guaranteed to obtain the global optimum without the assumption of complete recourse. This significantly generalizes the classic SDDP algorithms. We also characterize the iteration complexity of the proposed algorithms. In particular, for a ( T + 1 ) -stage stochastic MINLP satisfying L -exact Lipschitz regularization with d -dimensional state spaces, to obtain an ε -optimal root node solution, we prove that the number of iterations of the proposed deterministic sampling algorithm is upper bounded by O ( ( 2 L T ε ) d ) , and is lower bounded by O ( ( LT 4 ε ) d ) for the general case or by O ( ( LT 8 ε ) d / 2 - 1 ) for the convex case. This shows that the obtained complexity bounds are rather sharp. It also reveals that the iteration complexity depends polynomially on the number of stages. We further show that the iteration complexity depends linearly on T , if all the state spaces are finite sets, or if we seek a ( T ε ) -optimal solution when the state spaces are infinite sets, i.e. allowing the optimality gap to scale with T . To the best of our knowledge, this is the first work that reports global optimization algorithms as well as iteration complexity results for solving such a large class of multistage stochastic programs. The iteration complexity study resolves a conjecture by the late Prof. Shabbir Ahmed in the general setting of multistage stochastic mixed-integer optimization. In this paper, we study multistage stochastic mixed-integer nonlinear programs (MS-MINLP). This general class of problems encompasses, as important special cases, multistage stochastic convex optimization with non-Lipschitzian value functions and multistage stochastic mixed-integer linear optimization. We develop stochastic dual dynamic programming (SDDP) type algorithms with nested decomposition, deterministic sampling, and stochastic sampling. The key ingredient is a new type of cuts based on generalized conjugacy. Several interesting classes of MS-MINLP are identified, where the new algorithms are guaranteed to obtain the global optimum without the assumption of complete recourse. This significantly generalizes the classic SDDP algorithms. We also characterize the iteration complexity of the proposed algorithms. In particular, for a (T+1)-stage stochastic MINLP satisfying L-exact Lipschitz regularization with d-dimensional state spaces, to obtain an ε-optimal root node solution, we prove that the number of iterations of the proposed deterministic sampling algorithm is upper bounded by O((2LTε)d), and is lower bounded by O((LT4ε)d) for the general case or by O((LT8ε)d/2-1) for the convex case. This shows that the obtained complexity bounds are rather sharp. It also reveals that the iteration complexity depends polynomially on the number of stages. We further show that the iteration complexity depends linearly on T, if all the state spaces are finite sets, or if we seek a (Tε)-optimal solution when the state spaces are infinite sets, i.e. allowing the optimality gap to scale with T. To the best of our knowledge, this is the first work that reports global optimization algorithms as well as iteration complexity results for solving such a large class of multistage stochastic programs. The iteration complexity study resolves a conjecture by the late Prof. Shabbir Ahmed in the general setting of multistage stochastic mixed-integer optimization. In this paper, we study multistage stochastic mixed-integer nonlinear programs (MS-MINLP). This general class of problems encompasses, as important special cases, multistage stochastic convex optimization with non-Lipschitzian value functions and multistage stochastic mixed-integer linear optimization. We develop stochastic dual dynamic programming (SDDP) type algorithms with nested decomposition, deterministic sampling, and stochastic sampling. The key ingredient is a new type of cuts based on generalized conjugacy. Several interesting classes of MS-MINLP are identified, where the new algorithms are guaranteed to obtain the global optimum without the assumption of complete recourse. This significantly generalizes the classic SDDP algorithms. We also characterize the iteration complexity of the proposed algorithms. In particular, for a [Formula omitted]-stage stochastic MINLP satisfying L-exact Lipschitz regularization with d-dimensional state spaces, to obtain an [Formula omitted]-optimal root node solution, we prove that the number of iterations of the proposed deterministic sampling algorithm is upper bounded by [Formula omitted], and is lower bounded by [Formula omitted] for the general case or by [Formula omitted] for the convex case. This shows that the obtained complexity bounds are rather sharp. It also reveals that the iteration complexity depends polynomially on the number of stages. We further show that the iteration complexity depends linearly on T, if all the state spaces are finite sets, or if we seek a [Formula omitted]-optimal solution when the state spaces are infinite sets, i.e. allowing the optimality gap to scale with T. To the best of our knowledge, this is the first work that reports global optimization algorithms as well as iteration complexity results for solving such a large class of multistage stochastic programs. The iteration complexity study resolves a conjecture by the late Prof. Shabbir Ahmed in the general setting of multistage stochastic mixed-integer optimization. In this paper, we study multistage stochastic mixed-integer nonlinear programs (MS-MINLP). This general class of problems encompasses, as important special cases, multistage stochastic convex optimization with non-Lipschitzian value functions and multistage stochastic mixed-integer linear optimization. We develop stochastic dual dynamic programming (SDDP) type algorithms with nested decomposition, deterministic sampling, and stochastic sampling. The key ingredient is a new type of cuts based on generalized conjugacy. Several interesting classes of MS-MINLP are identified, where the new algorithms are guaranteed to obtain the global optimum without the assumption of complete recourse. This significantly generalizes the classic SDDP algorithms. We also characterize the iteration complexity of the proposed algorithms. In particular, for a $$(T+1)$$ ( T + 1 ) -stage stochastic MINLP satisfying L -exact Lipschitz regularization with d -dimensional state spaces, to obtain an $$\varepsilon $$ ε -optimal root node solution, we prove that the number of iterations of the proposed deterministic sampling algorithm is upper bounded by $${\mathcal {O}}((\frac{2LT}{\varepsilon })^d)$$ O ( ( 2 L T ε ) d ) , and is lower bounded by $${\mathcal {O}}((\frac{LT}{4\varepsilon })^d)$$ O ( ( LT 4 ε ) d ) for the general case or by $${\mathcal {O}}((\frac{LT}{8\varepsilon })^{d/2-1})$$ O ( ( LT 8 ε ) d / 2 - 1 ) for the convex case. This shows that the obtained complexity bounds are rather sharp. It also reveals that the iteration complexity depends polynomially on the number of stages. We further show that the iteration complexity depends linearly on T , if all the state spaces are finite sets, or if we seek a $$(T\varepsilon )$$ ( T ε ) -optimal solution when the state spaces are infinite sets, i.e. allowing the optimality gap to scale with T . To the best of our knowledge, this is the first work that reports global optimization algorithms as well as iteration complexity results for solving such a large class of multistage stochastic programs. The iteration complexity study resolves a conjecture by the late Prof. Shabbir Ahmed in the general setting of multistage stochastic mixed-integer optimization.
Audience	Academic
Author	Zhang, Shixuan Sun, Xu Andy
Author_xml	– sequence: 1 givenname: Shixuan surname: Zhang fullname: Zhang, Shixuan organization: H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology – sequence: 2 givenname: Xu Andy orcidid: 0000-0003-3917-9418 surname: Sun fullname: Sun, Xu Andy email: sunx@mit.edu organization: Sloan School of Management, Operations Research Center, Massachusetts Institute of Technology
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Snippet	In this paper, we study multistage stochastic mixed-integer nonlinear programs (MS-MINLP). This general class of problems encompasses, as important special...
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SubjectTerms	Algorithms Calculus of Variations and Optimal Control; Optimization Combinatorics Complexity Computational geometry Convexity Dynamic programming Full Length Paper Global optimization Investment analysis Mathematical and Computational Physics Mathematical Methods in Physics Mathematical optimization Mathematics Mathematics and Statistics Mathematics of Computing Mixed integer Numerical Analysis Optimization Regularization Sampling Theoretical
Title	Stochastic dual dynamic programming for multistage stochastic mixed-integer nonlinear optimization
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