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

Abstract

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 $$(T+1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>T</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -stage stochastic MINLP satisfying L -exact Lipschitz regularization with d -dimensional state spaces, to obtain an $$\varepsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ε</mml:mi> </mml:math> -optimal root node solution, we prove that the number of iterations of the proposed deterministic sampling algorithm is upper bounded by $${\mathcal {O}}((\frac{2LT}{\varepsilon })^d)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mfrac> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>L</mml:mi> <mml:mi>T</mml:mi> </mml:mrow> <mml:mi>ε</mml:mi> </mml:mfrac> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , and is lower bounded by $${\mathcal {O}}((\frac{LT}{4\varepsilon })^d)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mfrac> <mml:mrow> <mml:mi>LT</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>4</mml:mn> <mml:mi>ε</mml:mi> </mml:mrow> </mml:mfrac> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> for the general case or by $${\mathcal {O}}((\frac{LT}{8\varepsilon })^{d/2-1})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mfrac> <mml:mrow> <mml:mi>LT</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>8</mml:mn> <mml:mi>ε</mml:mi> </mml:mrow> </mml:mfrac> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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 $$(T\varepsilon )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>T</mml:mi> <mml:mi>ε</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -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.