Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.27579

A Unified Theory of Solvability for Stochastic Linear-Quadratic Control Problems I: The Homogeneous Case

Hongwei Mei
View original source ↗Read source document ↗Export metadata ↓

Abstract

The existing theory of stochastic linear-quadratic (LQ) optimal control problems focuses on open-loop and closed-loop solvability. For finite horizon problems, it is widely accepted that those two types of solvability are not equivalent in general, as demonstrated by explicit counterexamples. In contrast, they coincide in standard infinite-horizon formulations with constant coefficients under suitable stabilizability and integrability assumptions. This paper aims at challenging this well-established paradigm by developing a novel framework of closed-loop solvability, namely generalized closed-loop solvability, which removes the $L^2$-integrability constraint on the feedback strategies from the existing formulation. It will be proven that open-loop and generalized closed-loop solvability are equivalent and the classical closed-loop solvablity is a particular case only when the admissible feedback strategy is $L^2$-integrable over the time horizon. In other words, the previously established non-equivalence between open-loop and closed-loop solvability stems solely from the limitations of the classical formulation which requires the $L^2$-integrability of the admissible feedback strategies over the time-horizon. Therefore, this paper unifies the theory of the solvabilitiy for stochastic LQ control problems. Moreover, our results offer a novel insight to stochastic LQ control problems: open-loop/generalized closed-loop solvability is essentially equivalent to piecewisely local closed-loop solvability in the classical sense. The current paper is focused on homogeneous cases only and non-homogeneous cases will be studied in a subsequent paper.

Explore related subjects

math.OC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Hongwei Mei. 2026-10-20. A Unified Theory of Solvability for Stochastic Linear-Quadratic Control Problems I: The Homogeneous Case. https://arxiv.org/abs/2610.27579

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

ItsDEAL: Inexact two-level smoothing descent algorithms for weakly convex optimization

This paper addresses nonconvex optimization problems via a two-level smoothing framework, where the high-order Moreau envelope (HOME) provides a smooth approximation of weakly convex cost functions. We study the differentiability and weak smoothness of HOME. Building on the inexact two-level smoothing optimization framework (ItsOPT), we propose an inexact two-level smoothing descent algorithm (ItsDEAL) by: (i) approximately solving the proximal subproblem to obtain an inexact first-order oracle for HOME at the lower level; and (ii) applying an inexact first-order method at the upper level. We derive computable sufficient conditions for lower-level inexactness, enabling approximate high-order proximal-points to be certified without access to the exact proximal-point or HOME value. In particular, we study a parameter-free Hölderian inexact gradient method and an inexact descent method with Armijo line search, both exploiting the weak smoothness of HOME. We establish subsequential convergence under mild inexactness assumptions, and global and linear convergence under the Kurdyka--Łojasiewicz (KL) property. Preliminary experiments on ridge-regularized hinge-loss classification, a high-dimensional MCP-regularized classification model, and robust matrix completion illustrate the numerical behavior of the proposed methods and the role of the HOME order.

math.OC↗
arXiv2026

OFF Newton-type Methods for Unconstrained Optimization with Inexact Derivatives

We study objective-function-free (OFF) Newton-type methods for smooth unconstrained optimization with inexact gradients and Hessians. We propose two methods for first- and second-order stationarity, coupling explicit step-size rules with derivative-accuracy and linear-solver residual conditions to guarantee sufficient decrease without evaluating the objective function. The OFF regularized Newton method finds an $\varepsilon$-first-order stationary point in $\mathcal{O}(\varepsilon^{-2})$ iterations. Under strong convexity, a local Newton variant converges with \(Q\)-order at least $1+θ$, where \(θ\in (0, 1]\). For second-order stationarity, we combine regularized Newton steps with approximate eigenvectors associated with the minimum eigenvalue. With high probability, the resulting method finds an $(\varepsilon, \sqrt{L_h\varepsilon})$-second-order stationary point in $\mathcal{O}(\varepsilon^{-3/2})$ iterations, using a total of $\widetilde{\mathcal{O}}(\varepsilon^{-7/4})$ approximate gradient evaluations and Hessian-vector products. For finite-sum problems, we develop subsampling schemes that yield best-iterate convergence in expectation in the nonconvex first-order setting and local \(R\)-linear convergence with high probability in the strongly convex setting, followed by \(Q\)-order at least \(1+θ\) once full sampling is reached. The subsampled second-order method achieves a high-probability $\widetilde{\mathcal{O}}(\varepsilon^{-7/2})$ total sample-operation bound, counting component gradient evaluations and component Hessian-vector products.

math.OC↗
arXiv2026

An efficient mixed-integer linear programming formulation for solving influence diagrams

Influence diagrams represent decision-making problems with interdependencies between random events, decisions, and consequences. Traditionally, they have been solved using tailored algorithms that determine the expected utility-maximizing decision strategy. Towards this end, most recent state-of-the-art solution approaches convert them into mixed-integer linear programming (MILP) models that can be solved using off-the-shelf MILP solvers. Computationally, these approaches are most efficient for influence diagrams that represent periodic (or sequential) decision processes, which can be cast as partially observable Markov Decision Processes. However, they are inefficient for problems that lack this periodic structure or whose influence diagrams have large state spaces. The efficient MILP formulation presented in this paper is specifically designed for influence diagrams that are challenging for earlier MILP formulations. Additionally, this formulation can be adapted to optimize conditional value-at-risk and to incorporate chance and logical constraints, thereby retaining the modeling flexibility of MILP-based methods. Finally, computational experiments on a range of problems from the literature are reported to show that the proposed formulation is much more efficient than earlier MILP formulations, particularly in problems which cannot be modeled as partially observable Markov decision processes and which lead to influence diagrams whose decision or chance nodes may have large state spaces.

math.OC↗