(Communicated by Maria Josefa Canovas)
We propose a vector linear programming formulation for a non-stationary, finite-horizon Markov decision process with vector-valued rewards. Pareto efficient policies are shown to correspond to efficient solutions of the linear program, and vector linear programming theory allows us to fully characterize deterministic efficient policies. An algorithm for enumerating all efficient deterministic policies is presented then tested numerically in an engineering application.
(Communicated by Guanglu Zhou)
In this paper, we focus on the mixed-constrained semi-infinite programming and present an adaptive augmented Lagrangian filter method. Continuous infinite inequality constraints are smoothed and transformed into equivalent finite constraints in integral form, which are then combined with the objective function via adaptive parameters. Compared with the existing methods, the new approach is a penalty-function-free method that employs a filter instead of forcing sequences to make the optimality measure approach to zero. Our algorithm also includes a feasibility recovery phase to quickly detect infeasible problems. The global convergence is proved under some suitable conditions. Numerical results demonstrate that the proposed method is effective.
(Communicated by Jie Sun)
In this paper, we consider the 3-block linearly constrained difference-of-convex (DC) optimization problems:
$\min_{x,y,z}\ \{ \ f(x) + g (y) + h(z) \mid A x +By+Cz =b \ \},$ (1)
where
$f(x) = f_1(x)-f_2(x)~ ~\text{and}~~ g(y) = g_1(y)-g_2(y),$
$h:\mathbb{R}^{n_3}\rightarrow \mathbb{R}$ be a continuous differentiable convex function with Lipschitz continuous gradient, $A\in \mathbb{R}^{m\times n_1}$, $B\in \mathbb{R}^{m\times n_2}$ and $C\in\mathbb{R}^{m\times n_3}$ are given matrixes, $b\in \mathbb{R}^{m}$ is a vector, with $f_1:\mathbb{R}^{n_1}\rightarrow \mathbb{R}\cup\{+\infty\}$ and $g_1:\mathbb{R}^{n_2}\rightarrow \mathbb{R}\cup\{+\infty\}$ are proper closed convex functions, $f_2:\mathbb{R}^{n_1}\rightarrow \mathbb{R}\cup\{+\infty\}$ and $g_2:\mathbb{R}^{n_2}\rightarrow \mathbb{R}\cup\{+\infty\}$ are continuous convex functions. We propose a majorized Bregman ADMM to solve the DC problem (1). Compared with the classical Bregman ADMM, the majorized Bregman ADMM only requires solving convex subproblems at each iteration, rather than DC subproblems. We prove that the sequence generated by the proposed method converges to a critical point of the augmented Lagrangian function, under the assumption that the potential function satisfies the Kurdyka-Łojasiewicz (KŁ) property. Preliminary numerical experiments are conducted to support our theoretical analysis.
(Communicated by Lingchen Kong)
We study a complex Newton-based method with a correction step for homogeneous polynomial optimization on the complex unit sphere. In particular, its constrained stationary point meets the definition for being a US-eigenpair of the corresponding to symmetric complex tensor.
We analyze the local quadratic convergence rate of the complex Newton-based method with a correction step, provided that a sufficiently close initial point. As two concrete applications, we apply it to two tasks: computing the US-eigenpairs of symmetric complex tensors; and computing the geometric measure of entanglement of quantum multipartite pure states. We provide several numerical examples of our complex Newton-based method with a correction step, and observe that it performs fast and effective on these two tasks.
(Communicated by Fanwen Meng)
The split feasibility problem (SFP) has many important and wide applications, such as radiotherapy, image reconstruction, and signal processing. Let $C$ and $Q$ be nonempty closed convex sets in $\Re^{n}$ and $\Re^{m}$, respectively, and $A$ an $m \times n$ real matrix, this kind of problem is about finding
$x\in C, ~\mathrm{s.t.}~Ax\in Q,$ (1)
if such $x$ exists.
Recent researches on the SFP have been deeply integrated with inertial acceleration method, and some scholars have discovered that these techniques can speed up the process of first-order optimization algorithms. However, the most of them limit the range of inertial factors and cause the sequence $\|x^{k}-z\|(z\in S)$ no longer monotonically non-increasing. The alternating inertial method, expressed as
$w^{k}=\begin{cases}x^{k},&\text{if ~}k\text{~is even},\\ x^{k}+\theta _{k}( x^{k}-x^{k-1}) ,&{\text{if ~}k\text{~is odd}},\end{cases}$ (2)
is a appropriate improvement strategy, which retains the acceleration effect of inertial method while alleviating the degree of oscillation.
In this paper, we propose three KM-CQ-like alternating inertial algorithms that all expand the range of inertial factors. We use projections onto general closed convex sets in the first version, which exploring the acceleration effects of alternating inertial technique preliminarily. The second version simplifies the computation of procedure by relaxing projections onto half-spaces. The last version further optimizes the previous methods by incorporating double projection technique.
Our specific contributions in this paper are summarized as follows.
(i) The proposed three KM-CQ-like algorithms incorporate alternating inertial steps allowing them to improve the convergence of the algorithms without inertial steps, which improve the range of values of inertial factors and simplify the form of limiting condition.
(ii) The second algorithm also add relaxation effects that allows it to accelerate the convergence of the first algorithm and simplify the calculation of projections. On the other hand, we can see it is faster in Example 4.2.
(iii) The last algorithm use double projections that are generated by half-spaces in the previous iteration and the current iteration. Similar to the expected conclusions, it converges faster than the second algorithm in Example 4.2.
(iv) The convergence of the iterative sequences generated by the proposed algorithms is established, with the monotonicity of sequence $\|{x}^{2k}-z\|(z\in S)$, which alleviates the oscillation of the iterative points.
(v) The performance and advantages of the algorithms proposed in this paper are confirmed by two applications in a simple the SFP and signal processing. On the other hand, the increase of dimension will make the advantages of our algorithms more obvious.
Under simple parameter settings, these algorithms restore the monotonicity of $\|x^{2k}-z\|$ and achieve the convergence of the algorithm. Experiments demonstrate the feasibility of each algorithm in practical applications and the effectiveness of the acceleration procedure.
(Communicated by Kok Lay Teo)
In this study, we examine the linear quadratic (LQ) optimal control problem for large-scale interconnected systems, with a focus on designing sparse static state-feedback controllers. Traditional LQ control methods often result in dense feedback matrices that require global information sharing among all subsystems. While such dense designs may be optimal in terms of performance, they are often impractical in distributed or resource-constrained environments due to excessive communication and implementation costs. Consequently, this paper seeks to develop a framework that simultaneously promotes sparsity in the controller while maintaining an acceptable level of system performance degradation.
To achieve this, we formulate a constrained optimization problem where the primary objective is to minimize the number of nonzero entries in the feedback matrix $K$, expressed via the nonconvex and discontinuous $\ell_0-$norm. This sparsity-promoting goal is constrained by two critical system-level considerations: (i) the cost associated with the closed-loop system's response to disturbances must not exceed a predefined threshold above the optimal LQ cost, and (ii) the stability of the closed-loop system must be preserved via a Lyapunov-type equation. The cost of the closed-loop system is defined as the trace of a controllability Gramian $P$, which satisfies the Lyapunov-type equation involving the feedback matrix $K$. The constraint on system performance introduces a tradeoff parameter $\gamma > 0$, which allows the system designer to control the degree of allowable performance loss relative to the optimal centralized LQ controller. This parameter directly impacts the achievable sparsity of the feedback matrix: larger values of $\gamma$ encourage sparser solutions at the expense of higher communication cost, and vice versa.
Given the nonconvexity introduced by the $\ell_0-$norm and the complexity of the Lyapunov-based equality constraint, solving this problem directly is computationally challenging. To address this, we adopt an augmented Lagrangian framework that transforms the constrained optimization into an unconstrained minimization problem. The resulting augmented Lagrangian function comprises three components, namely the $\ell_0-$norm of the feedback matrix, an indicator function representing the constraint on the cost function, and a matrices quadratic equation associated with the Lyapunov-tpe equation, involving a matrix of Lagrange multipliers. This formulation separates the objective into two nonsmooth components, corresponding to the sparsity and cost constraints, coupled through a smooth term arising from the system dynamics and Lyapunov relation. To solve the augmented problem, we propose a Proximal Linearized Minimization (PLM) method tailored to the two-block structure of the augmented objective function. Each iteration alternates between updating $K$ and $P$ using a proximal-gradient step designed for nonsmooth and nonconvex functions. The convergence of our method is analyzed using the Kurdyka-Łojasiewicz (KŁ) inequality, a powerful framework for analyzing nonconvex and nonsmooth optimization algorithms. Under mild regularity conditions, we prove that the PLM algorithm converges to a critical point of the augmented objective function. This analysis guarantees stability and consistency of the proposed framework, making it suitable for control system applications where reliable performance is essential.
We demonstrate the effectiveness of our approach through a series of numerical experiments on interconnected systems. The results show that our method can identify sparse feedback matrices that achieve comparable closed-loop performance to traditional optimal control designs, but with significantly reduced controller complexity and communication requirements.
(Communicated by Jie Sun)
This paper investigates a class of optimal control problems characterized by piecewise constant time delays, which frequently arise in engineering, biological, and networked systems. These problems pose significant challenges due to the discontinuous nature of the delays and the resulting complexity in computing gradients required for optimization. Within the control parameterization framework, the original infinite-dimensional problem is transformed into a finite-dimensional nonlinear programming problem, enabling the application of gradient-based optimization techniques. However, the traditional variational method for computing gradients of cost and constraint functions becomes increasingly inefficient as the control discretization is refined. Although the co-state method is well known for its computational efficiency in delay-free optimal control problems, its application to systems with piecewise constant time delays has been hindered by the absence of an explicit co-state system. In this work, we rigorously derive the co-state system corresponding to such problems, facilitating efficient and accurate gradient computation. Building on this derivation, we propose a computational framework that significantly accelerates the solution process without compromising accuracy. Numerical experiments demonstrate that the co-state method offers substantial improvements in computational efficiency over the variational approach, particularly in cases involving fine control discretization.
(Communicated by Robert Csetnek )
In this paper, a new approach to solving the considered optimization problem is proposed for both differential inclusions (DFIs) and inequality constraints. The problem is reduced to a problem with one DFI defined by the intersection of two set-valued mappings. The optimality conditions for such a problem are expressed as the sum of two locally adjoint mappings (LAMs). In turn, this also requires calculating the argmaximum sets and subdifferentials of Hamiltonian functions. Then it is natural that LAM can be computed for a set-valued mapping generated by a system of inequality constraints. Finally, we consider some applications in a linear optimal control problem with polyhedral constraints and a constraint given by a convex cone.
(Communicated by Jie Sun)
This study develops a data-driven distributionally robust two-stage stochastic programming model with second-order stochastic dominance (SSD) constraints. Uncertainty is modeled within a 1-Wasserstein ball centered at an empirical distribution. Two cases are analyzed based on how uncertainty affects the second stage. In the first case, where uncertainty impacts only the objective
function, the problem can be reformulated as a convex program and further simplified into a tractable conic program. In the second case, where uncertainty influences constraints, the problem is generally NP-hard, as feasibility verification educes to a norm maximization problem over a polytope. However, under specific conditions, it remains tractable. Numerical experiments on a supply chain problem compare the proposed method with Sample Average Approximation (SAA). Results show that for large sample sizes, the proposed approach exhibits greater robustness and stability against sample fluctuations.
(Communicated by Jie Sun)
This paper presents a novel multi-objective location-routing problem with time windows for the effective management of multimodal transportation networks at two levels: the upper level addresses multimodal transportation issues, whereas the lower one focuses on the location-routing aspects. In our model, two objectives are included referring to maximizing customer satisfaction and minimizing total costs, aggregating transportation, transit, and location costs. To solve the problem efficiently, we introduce a new approach based on a squirrel search algorithm using three crossover and four mutation operators designed to handle Pareto optimality in multi-objective optimization. Besides, we present a greedy clustering algorithm combined with Metropolis criterion to generate high-quality initial solutions and accelerate convergence. We evaluate the performance of proposed approach using various scale instances in multimodal transportation networks. Our experimental results suggest the efficiency and scalability of our approach.
