This paper introduces a new numerical method for solving nonlinear optimal control problems. In each iteration, the method linearizes the system and constraints based on the results from the previous iteration. Simultaneously, the objective function is approximated by its second-order Taylor expansion with respect to the control and state vectors, resulting in a linear-quadratic optimal control sub-problem. In solving the sub-problem, both the state and control vectors are approximated using Chebyshev series, where the coefficients of the Chebyshev polynomials are to be optimized. Furthermore, we approximate the known coefficient functions of the dynamic system and constraints using Chebyshev series. By the properties of Chebyshev polynomials, the sub-problem is finally transformed into a quadratic programming problem. We solve the dual problem of this optimization problem since it admits an exact solution. The effectiveness of the proposed method is demonstrated by examining three distinct types of optimal control problems. The numerical results conclusively illustrate that our approach outperforms the conventional iterative Chebyshev approximation method in terms of computation efficiency.
In this paper, the solvability and stability for a class of history-dependent quasi-variational hemivariational inequalities are investigated, where the constraint set of the considered problem depends on its solution. A unique solvability result to the history-dependent quasi-variational hemivariational inequality is established by applying a fixed point argument about history-dependent operators and the Gronwall inequality in a space of continuous functions. In addition, when the data of the history-dependent quasivariational hemivariational inequality are perturbed, sufficient conditions are given to guarantee that the solution sequence of the perturbed problem converges to the unique solution of the original problem.
The alternating direction method of multipliers (ADMM) has demonstrated its efficiency and well-understood convergence properties when applied to minimization problems where the objective function is the sum of two nonconvex separable functions and the constraint is linear. However, the requirement for global Lipschitz continuity of the gradient of differentiable functions, which is often impractical in nonconvex optimization problems, restricts its applicability across various domains. Recently, a novel Bregman ADMM has been introduced for two-block nonconvex optimization problems with linear constraints. This new Bregman ADMM not only removes the need for global Lipschitz continuity of the gradient, making it suitable for a broader range of practical problems, but also ensures that it can reduce to the classical ADMM in specific cases. Building on this Bregman ADMM, we address multi-block nonconvex separable optimization problems with linear constraints. We demonstrate that any cluster point of the iterative sequence generated by Bregman ADMM is a critical point, provided that the associated function satisfies the Kurdyka-Lojasiewicz inequality. Additionally, we present sufficient conditions to ensure both the convergence and convergence rate of the algorithm.
The Peaceman-Rachford splitting method (PRSM) can effectively solve two-block separable convex optimization problems with linear constraints. However, extending it directly to multi-block separable convex optimization problems lacks of convergence guarantees. To address this limitation, we introduce a generalized PRSM with an indefinite proximal term and a substitution step (GPRSM-S). The global convergence and the iteration complexity are analyzed by using variational inequality theory under mild assumptions. Finally, numerical experiments on the robust Principal Component Analysis (PCA) problem demonstrate that GPRSM-S has higher efficiency compared to previous approaches.
This paper investigates an effective branch-and-bound algorithm for solving the generalized linear fractional multiplicative programming (GLFMP) problem. Initially, leveraging the structure of GLFMP, some variables are introduced to transform it into an equivalent problem. Subsequently, bidirectional linear relaxation is applied to the constraint functions of the equivalent problem, resulting in its linear relaxation problem, which is embedded into the branch-and-bound framework. Furthermore, incorporating tailored region reduction technique, we propose an output space branch-and-bound optimization algorithm. Additionally, the convergence and complexity of the algorithm are analyzed. Finally, the feasibility and effectiveness of the algorithm are validated using GLFMP instances ranging from specific to general cases.
We investigate the distributed optimization problems, where the goal is to seek the global minimizer of the sum of smooth and strongly convex local objective functions. All local objective functions are nested on a strongly connected directed graph. Using the row-stochastic matrices and the Nesterov's momentum acceleration technique, we propose a unified and accelerated version of distributed gradient methods for directed graphs (UADM). In contrast to the methods using the column-stochastic matrices, UADM does not require each agent to know its out-degree, which leads to certain practical advantages in broadcast-based interaction scenarios. UADM unifies some well known distributed gradient methods, including Li-Row, FROST and FROZEN, as its special cases. Theoretical analysis shows that UADM has a linear convergence rate when the step-sizes and the momentum coefficients satisfy some upper bounds. Some numerical experiments are presented to verify that our method can achieve accelerated convergence in comparison with some existing distributed algorithms.
In this paper, we concentrate on a broad class of large-scale nonconvex and nonsmooth optimization problems. We first propose a novel inertial stochastic Bregman proximal alternating linearized minimization algorithm (TiSBPALM), which employs variance-reduced stochastic gradient estimators. Subsequently, under the assumption that the objective function satisfies the Kurdyka-Lojasiewicz property and certain conditions on the parameters are imposed, we prove that the sequence generated by our algorithm converges to a critical point in expectation. Additionally, we provide the convergence rate for the iteration sequence. Finally, we conduct numerical experiments on sparse nonnegative matrix factorization and blind image-deblurring to verify the effectiveness of our proposed algorithms.
In this paper, we study the trust region subproblem with a linear cut over the complex domain (TL). We first reveal its hidden convexity property by proving that it is equivalent to a convex programming problem. Then we present a linear-time algorithm approximation scheme (in terms of the number of nonzero entries) for solving (TL). It mainly employs the eigenvalue approximation technique and Nesterov's accelerated gradient descent algorithm. We show the efficiency of the algorithm by comparing it with CVX solver in the numerical experiment.
Solving the distributional worst-case in the distributionally robust optimization problem is equivalent to finding the projection onto the intersection of simplex and singly linear inequality constraint. This projection is a key component in the design of efficient first-order algorithms. This paper focuses on developing efficient algorithms for computing the projection onto the intersection of simplex and singly linear inequality constraint. Based on the Lagrangian duality theory, the studied projection can be obtained by solving a univariate nonsmooth equation. We employ an algorithm called LRSA, which leverages the Lagrangian duality approach and the secant method to compute this projection. In this algorithm, a modified secant method is specifically designed to solve the piecewise linear equation. Additionally, due to semismoothness of the resulting equation, the semismooth Newton (SSN) method is a natural choice for solving it. Numerical experiments demonstrate that LRSA outperforms SSN algorithm and the state-of-the-art optimization solver called Gurobi. Moreover, we derive explicit formulas for the generalized HS-Jacobian of the projection, which are essential for designing second-order nonsmooth Newton algorithms.
