arXiv subjects

Haiming Song

Publications and source records attributed to Haiming Song.

4 recordsLinked to original sources

arXiv2026

Faster Stochastic ADMM for Nonsmooth Composite Convex Optimization in Hilbert Space

In this paper, a stochastic alternating direction method of multipliers (ADMM) is proposed for a class of nonsmooth composite and stochastic convex optimization problems in Hilbert space, motivated by optimization problems constrained by partial differential equation (PDE) with random coefficients. We prove the strong convergence of the proposed ADMM algorithm in the strongly convex case, and show the faster nonergodic convergence rates in terms of functional values and feasibility violation for both strongly convex and general convex cases. We demonstrate the application of the proposed method to solve certain model problems, along with its associated probability bound of large deviation. Some preliminary numerical results illustrate the efficiency of our method.

math.OC↗

arXiv2026

An Inexact Alternating Direction Method of Multipliers for Constrained Parabolic Optimal Distributed Control Problems

Solving parabolic optimal control problems can be inherently challenging in the field of science and engineering, especially with constraints on the nonsmooth distributed control. Motivated by the extensive applicability of the alternating direction method of multipliers, in this paper we develop a novel inexact algorithmic framework for parabolic optimal distributed control problems with control constraints. By decoupling the control constraint and possible nonsmooth objective from the optimal control problem, our aim is to efficiently solve the subproblem constrained by the parabolic state equation, for which computing a sufficiently accurate numerical solution can be prohibitively expensive. Given this high computational cost, we consider that it may not always be justifiable to compute a highly accurate solution of the subproblem at every iteration. Hence, we propose an inexact strategy for solving the parabolic equation constrained subproblem. Under mild and flexible conditions on the parameters, we prove global convergence and a linear convergence rate for the resulting algorithmic framework. In practice, our inexact algorithmic framework is easily implementable with applicable nested iterations. Numerical experiments are performed on different cases, and the results demonstrate the validity and trustworthy performance of the proposed methods.

math.OC↗

arXiv2025

Computational Methods and Verification Theorem for Portfolio-Consumption Optimization under Exponential O-U Dynamics

In this paper, we focus on the problem of optimal portfolio-consumption policies in a multi-asset financial market, where the n risky assets follow Exponential Ornstein-Uhlenbeck processes, along with one risk-free bond. The investor's preferences are modeled using Constant Relative Risk Aversion utility with state-dependent stochastic discounting. The problem can be formulated as a high-dimensional stochastic optimal control problem, wherein the associated value function satisfies a Hamilton-Jacobi-Bellman (HJB) equation, which constitutes a necessary condition for optimality. We apply a variable separation technique to transform the HJB equation to a system of ordinary differential equations (ODEs). Then a class of hybrid numerical approaches that integrate exponential Rosenbrock-type methods with Runge-Kutta methods is proposed to solve the ODE system. More importantly, we establish a rigorous verification theorem that provides sufficient conditions for the existence of value function and admissible optimal control, which can be verified numerically. A series of experiments are performed, demonstrating that our proposed method outperforms the conventional grid-based method in both accuracy and computational cost. Furthermore, the numerically derived optimal policy achieves superior performance over all other considered admissible policies.

math.OC↗

arXiv2023

A splitting algorithm for constrained optimization problems with parabolic equations

In this paper, an efficient parallel splitting method is proposed for the optimal control problem with parabolic equation constraints. The linear finite element is used to approximate the state variable and the control variable in spatial direction. And the Crank-Nicolson scheme is applied to discretize the constraint equation in temporal direction. For consistency, the trapezoidal rule and midpoint rule are used to approximate the integrals with respect to the state variable and the control variable of the objective function in temporal direction, respectively. Based on the separable structure of the resulting coupled discretized optimization system, a full Jacobian decomposition method with correction is adopted to solve the decoupled subsystems in parallel, which improves the computational efficiency significantly. Moreover, the global convergence estimate is established using the discretization error by the finite element and the iteration error by the full Jacobian decomposition method with correction. Finally, numerical simulations are carried out to verify the efficiency of the proposed method.

math.OC↗