Adriana Washington | Searcharxiv

arXiv2026

Stochastic Quadratic Dynamic Programming

We introduce an algorithm called SQDP (Stochastic Quadratic Dynamic Programming) to solve some multistage stochastic optimization problems having strongly convex recourse functions. The algorithm extends the classical Stochastic Dual Dynamic Programming (SDDP) method replacing affine cuts by quadratic cuts. We provide conditions ensuring strong convexity of the recourse functions and prove the convergence of SQDP. In the special case of a single stage deterministic problem, we call QCSC (Quadratic Cuts for Strongly Convex optimization) the method and prove its complexity. Numerical experiments illustrate the performance and correctness of SQDP, with SQDP being much quicker than SDDP for large values of the constants of strong convexity both for a multistage problem and a two-stage assembly recourse model. We also present the results of numerical experiments on deterministic problems where QCSC is much quicker than several popular competing optimizers for solving 6 strongly convex optimization problems from the literature.

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arXiv2024

Bundle methods with quadratic cuts for deterministic and stochastic strongly convex optimization problems

We introduce two new methods for deterministic convex optimization problems: QCC (Quadratic Cuts for Convex optimization) and QB (Quadratic Bundle method). We prove the complexity of these methods for composite optimization problems which are the sum of a convex function $\tilde h$ and of a strongly convex function $\tilde f$ with parameter $μ$. These methods use as building blocks quadratic approximations of the strongly convex function $\tilde f$ where the quadratic terms are of form $\fracμ{2}\|\cdot-x_i\|^2$ for trial points $x_i$ computed along iterations (when $μ=0$ the building blocks are linear approximations). We extend the idea of using quadratic approximations to pieces of the objective for some multistage stochastic optimization problems which have strongly convex recourse functions that we approximate as a maximum of quadratic cuts. We call DASC (Dynamic Approximation for Strongly Convex optimzation) the corresponding optimization method. When the cuts are linear, the method boils down to the popular Stochastic Dual Dynamic Programming (SDDP) method. We provide conditions ensuring strong convexity of the recourse functions and prove the convergence of DASC. Numerical experiments illustrate the performance and correctness of DASC, with DASC being much quicker than SDDP for large values of the constants of strong convexity.

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