arXiv subjects

Haisen Zhang

Publications and source records attributed to Haisen Zhang.

14 recordsLinked to original sources

arXiv2026

Verifiable constraint qualifications for infinite-dimensional optimization problems and their applications to optimal control problems

This paper employs a finite codimensionality condition to establish an enhanced Fritz John condition for general constrained nonlinear infinite-dimensional optimization problems. In applications, this approach provides a new, unified framework for deriving first-order necessary conditions across a broad class of optimal control problems involving both deterministic and stochastic systems. Compared to existing constraint qualifications, our finite codimensionality condition, which is equivalent to the validity of certain \textit{a priori} estimates, yields a direct and analytically tractable verification method for each application. Furthermore, the core ideas of this method can be further extended to investigate the KKT conditions for infinite-dimensional optimization problems.

math.OC↗

arXiv2026

Convergence Analysis of Newton Methods for Nonlinear Optimal Control Problems

The main purpose of this paper is to establish the local quadratic convergence of Newton's method for nonlinear optimal control problems. Under suitable smoothness assumptions, the cost functional is first shown to be twice G{â}teaux differentiable with respect to the control. An explicit operator representation of its second derivative is derived, and its continuity in different control spaces is examined. The second-order coercivity condition in $L^2(0,T;\maathbb{R}^m)$ is then shown to be equivalent to the existence of a strongly regular solution to an associated matrix Riccati differential equation, thereby expressing an infinite-dimensional quadratic-form condition in terms of the solvability of a matrix differential equation. On this basis, local quadratic convergence of Newton's method in $L^\infty(0,T;\maathbb{R}^m)$ is established using linear-quadratic optimal control theory. Under suitable structural assumptions, local quadratic convergence in $L^2(0,T;\maathbb{R}^m)$ is also established. Explicit estimates of the corresponding convergence neighborhoods are derived, and the theoretical results are illustrated by numerical examples.

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arXiv2026

Numerical Integration at the Intersection of Probability Theory, Number Theory and Dynamical Systems

This paper is devoted to the study of uniformly distributed (u.d.) sequences on general non-compact topological spaces. Convergence-determining classes and the equivalent characterizations of u.d. sequences on $T_4$ spaces are discussed. For Polish spaces endowed with regular probability measures, the existence of a countable convergence-determining class for u.d. sequences is proved by virtue of the tightness of Borel probability measures. It is shown that, under the condition that a countable convergence-determining class exists, both independent identically distributed (i.i.d.) randomly sampled sequences and orbit sequences of ergodic transformations are u.d.\ with full measure.

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arXiv2024

Strong Convergence of Relaxed Inertial Inexact Progressive Hedging Algorithm for Multi-stage Stochastic Variational Inequality Problems

A Halpern-type relaxed inertial inexact progressive hedging algorithm (PHA) is proposed for solving multi-stage stochastic variational inequalities in general probability spaces. The subproblems in this algorithm are allowed to be calculated inexactly. It is found that the Halpern-type relaxed inertial inexact PHA is closely related to the Halpern-type relaxed inertial inexact proximal point algorithm (PPA). The strong convergence of the Halpern-type relaxed inertial inexact PHA is proved under appropriate conditions. Some numerical examples are given to indicate that the over-relaxed parameter and the inertial term can accelerate the convergence of the algorithm.

math.OC↗

arXiv2023

A prediction-correction ADMM for multistage stochastic variational inequalities

The multistage stochastic variational inequality is reformulated into a variational inequality with separable structure through introducing a new variable. The prediction-correction ADMM which was originally proposed in [B.-S. He, L.-Z. Liao and M.-J. Qian, J. Comput. Math., 24 (2006), 693--710] for solving deterministic variational inequalities in finite dimensional spaces is adapted to solve the multistage stochastic variational inequality. Weak convergence of the sequence generated by that algorithm is proved under the conditions of monotonicity and Lipschitz continuity. When the sample space is a finite set, the corresponding multistage stochastic variational inequality is defined on a finite dimensional Hilbert space and the strong convergence of the sequence naturally holds true. A numerical example in that case is given to show the efficiency of the algorithm.

math.OC↗

arXiv2023

Lagrangian dual method for solving stochastic linear quadratic optimal control problems with terminal state constraints

A stochastic linear quadratic (LQ) optimal control problem with a pointwise linear equality constraint on the terminal state is considered. A strong Lagrangian duality theorem is proved under a uniform convexity condition on the cost functional and a surjectivity condition on the linear constraint mapping. Based on the Lagrangian duality, two approaches are proposed to solve the constrained stochastic LQ problem. First, a theoretical method is given to construct the closed-form solution by the strong duality. Second, an iterative algorithm, called augmented Lagrangian method (ALM), is proposed. The strong convergence of the iterative sequence generated by ALM is proved. In addition, some sufficient conditions for the surjectivity of the constraint mapping are obtained.

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arXiv2020

Deep Learning based Radio Resource Management in NOMA Networks: User Association, Subchannel and Power Allocation

With the rapid development of future wireless communication, the combination of NOMA technology and millimeter-wave(mmWave) technology has become a research hotspot. The application of NOMA in mmWave heterogeneous networks can meet the diverse needs of users in different applications and scenarios in future communications. In this paper, we propose a machine learning framework to deal with the user association, subchannel and power allocation problems in such a complex scenario. We focus on maximizing the energy efficiency (EE) of the system under the constraints of quality of service (QoS), interference limitation, and power limitation. Specifically, user association is solved through the Lagrange dual decomposition method, while semi-supervised learning and deep neural network (DNN) are used for the subchannel and power allocation, respectively. In particular, unlabeled samples are introduced to improve approximation and generalization ability for subchannel allocation. The simulation indicates that the proposed scheme can achieve higher EE with lower complexity.

cs.IT↗

arXiv2020

Energy Efficient User Clustering, Hybrid Precoding and Power Optimization in Terahertz MIMO-NOMA Systems

Terahertz (THz) band communication has been widely studied to meet the future demand for ultra-high capacity. In addition, multi-input multi-output (MIMO) technique and non-orthogonal multiple access (NOMA) technique with multi-antenna also enable the network to carry more users and provide multiplexing gain. In this paper, we study the maximization of energy efficiency (EE) problem in THz-NOMA-MIMO systems for the first time. And the original optimization problem is divided into user clustering, hybrid precoding and power optimization. Based on channel correlation characteristics, a fast convergence scheme for user clustering in THz-NOMA-MIMO system using enhanced K-means machine learning algorithm is proposed. Considering the power consumption and implementation complexity, the hybrid precoding scheme based on the sub-connection structure is adopted. Considering the fronthaul link capacity constraint, we design a distributed alternating direction method of multipliers (ADMM) algorithm for power allocation to maximize the EE of THz-NOMA cache-enabled system with imperfect successive interference cancellation (SIC). The simulation results show that the proposed user clustering scheme can achieve faster convergence and higher EE, the design of the hybrid precoding of the sub-connection structure can achieve lower power consumption and power optimization can achieve a higher EE for the THz cache-enabled network.

eess.SP↗

arXiv2018

Second Order Optimality Conditions for Optimal Control Problems of Stochastic Evolution Equations

In this paper, we establish some second order necessary/sufficient optimality conditions for optimal control problems of stochastic evolution equations in infinite dimensions. The control acts on both the drift and diffusion terms and the control region is convex. The concepts of relaxed and $V$-transposition solutions (introduced in our previous works) to operator-valued backward stochastic evolution equations are employed to derive these optimality conditions. The correction part of the second order adjoint equation, which does not appear in the (first order) Pontryagin-type stochastic maximum principle, plays a fundamental role in our second order optimality conditions.

math.OC↗

arXiv2016

First and second order necessary conditions for stochastic optimal controls

The main purpose of this paper is to establish the first and second order necessary optimality conditions for stochastic optimal controls using the classical variational analysis approach. The control system is governed by a stochastic differential equation, in which both drift and diffusion terms may contain the control variable and the set of controls is allowed to be nonconvex. Only one adjoint equation is introduced to derive the first order necessary condition; while only two adjoint equations are needed to state the second order necessary conditions for stochastic optimal controls.

math.OC↗

arXiv2016

Optimal control problems of forward-backward stochastic Volterra integral equations with closed control regions

Optimal control problems of forward-backward stochastic Volterra integral equations (FBSVIEs, in short) with closed control regions are formulated and studied. Instead of using spike variation method as one may imagine, here we turn to treat the non-convexity of the control regions by borrowing some tools in set-valued analysis and adapting them into our stochastic control systems. A duality principle between linear backward stochastic Volterra integral equations and linear stochastic Fredholm-Volterra integral equations with conditional expectation are derived, which extends and improves the corresponding results in [25], [30]. Some first order necessary optimality conditions for optimal controls of FBSVIEs are established. In contrast with existed common routines to treat the non-convexity of stochastic control problems, here only one adjoint system and one-order differentiability requirements of the coefficients are needed.

math.OC↗

arXiv2015

Pointwise second-order necessary conditions for stochastic optimal controls, Part II: The general case

This paper is the second part of our series of work to establish pointwise second-order necessary conditions for stochastic optimal controls. In this part, we consider the general cases, i.e., the control region is allowed to be nonconvex, and the control variable enters into both the drift and the diffusion terms of the control systems. By introducing four variational equations and four adjoint equations, we obtain the desired necessary conditions for stochastic singular optimal controls in the sense of Pontryagin-type maximum principle.

math.OC↗

arXiv2014

Pointwise second-order necessary conditions for stochastic optimal controls, Part I: The case of convex control constraint

This paper is the first part of our series work to establish pointwise second-order necessary conditions for stochastic optimal controls. In this part, both drift and diffusion terms may contain the control variable but the control region is assumed to be convex. Under some assumptions in terms of Malliavin calculus, we establish the desired necessary condition for stochastic singular optimal controls in the classical sense.

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arXiv2014

Some results on pointwise second-order necessary conditions for stochastic optimal controls

The purpose of this paper is to derive some pointwise second-order necessary conditions for stochastic optimal controls in the general case that the control variable enters into both the drift and the diffusion terms. When the control region is convex, a pointwise second-order necessary condition for stochastic singular optimal controls in the classical sense is established; while when the control region is allowed to be nonconvex, we obtain a pointwise second-order necessary condition for stochastic singular optimal controls in the sense of Pontryagin-type maximum principle. It is found that, quite different from the first-order necessary conditions, the correction part of the solution to the second-order adjoint equation appears in the pointwise second-order necessary conditions whenever the diffusion term depends on the control variable, even if the control region is convex.

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