As an effective acceleration technique, multi-step inertia has attracted increasing attention in the development of first-order methods. In this paper, we propose a multi-step inertial generalized Peaceman-Rachford splitting method (abbreviated as MIGPRSM) for solving a family of separable convex programming problems subject to linear constraints. The involved subproblems are linearized by tailored proximal terms, which could be solved possibly easier than that without employing proximal terms. The global convergence and sublinear convergence rate of MIGPRSM are analysed by variational characterization for both the saddle point of the problem and the iterative sequence. Numerical experiments on LASSO and low patch rank image decomposition problems are performed to verify the efficiency of our proposed method.
This paper investigates first-order optimization algorithms derived by discretizing two high-resolution ordinary differential equations. For sufficiently smooth objective functions, we demonstrate that acceleration can be achieved via Runge-Kutta integrators. Specifically, for convex objective functions f satisfying the assumptions of Lipschitz continuous gradients and (s+2) -th order differentiability, we prove that the objective function converges to its minimum value with a rate of O(N-2ss+1) , where s denotes the order of the Runge-Kutta integrator. For strongly convex objective functions f, under the same assumptions, we further prove that the proposed algorithms achieve a faster convergence rate than the algorithms generated by symplectic Euler discretization. Numerical experiments validate the effectiveness and superiority of the proposed algorithms through comparisons with classical benchmark methods, including Nesterov's accelerated gradient method and the symplectic Euler scheme.
In this paper, a steepest descent method (SDM) with a non-monotone line search strategy is developed to solve the robust counterpart of uncertain multiobjective optimization problems (UMOPs) under finite uncertainty. The uncertainty in the objective functions is addressed using an objective-wise maximum-type robust reformulation, which converts the original uncertain problem into a deterministic form. To solve this robust deterministic problem, a descent direction is computed by solving an auxiliary subproblem at each iteration. Subsequently, a non-monotone line search technique is applied to determine an appropriate step length, allowing flexibility in the acceptance of trial points and improving convergence behaviour. Two variants of non-monotonicity, maximum-type and average-type, are considered, and corresponding algorithms are formulated. The convergence of the proposed methods is established under standard assumptions. Finally, the effectiveness of both non-monotone algorithms is validated through a set of numerical experiments and compared with the existing SDM with monotone line search (an Armijo-type inexact line search technique) developed for UMOPs, as well as with an SDM equipped with non-monotone line search designed for multiobjective optimization problems. The comparison is carried out using performance profiles.
In this paper, we study a class of nonconvex composite optimization problems arising in data sciences, where the objective function consists of a nonconvex regularization term and a nonconvex beyond-quadratic loss function. Unfortunately, a direct application of the popular alternating direction method of multipliers (ADMM) to such problems is often hindered by the nonquadratic loss term, which typically prevents a closed-form solution and thus limits efficiency in large-scale settings. To overcome this challenge, we propose a practical implementation of ADMM based on the majorization-minimization strategy, which efficiently alleviates the difficulties associated with the nonquadratic term. Under mild assumptions, we show that the sequence generated by our algorithm converges to a critical point with the help of the Kurdyka-& Lstrok;ojasiewicz property. As an interesting application, we introduce a new & ell; (1) / & ell;(2) -norm regularized model with a Tukey biweight loss function tailored for problems involving non-Gaussian noise. Numerical experiments on some synthetic datasets demonstrate the promising performance of the proposed algorithm.
In this paper, we establish the connections between the solutions of some classes of vector-type variational control inequalities, denoted by (V CI) and (WV CI), and (local, weak) quasi-efficient solutions of the associated multiobjective optimization problem, denoted by (P). In this regard, we use local (strictly) approximately star-shaped and/or local approximately pseudo-convex integral functionals. In addition, some illustrative examples are also presented to verify the statements and established theoretical results.
The risk-sensitive first exit time stochastic zero-sum game problem for pure jump processes on a general state space with state-dependent discount factors is analysed. Under minimal assumptions, we prove the existence and uniqueness of the value of the game over the history-dependent policy space for bounded cost function and transition rates. We propose a value iteration algorithm and establish its convergence on a general state space to approximate the value of the game. For countable state space, we propose a new policy iteration algorithm and prove its convergence to the Saddle point equilibrium. Finally, we present two examples with bounded transition and cost rates to illustrate our convergence results corresponding to the general Borel and countable state space settings.
We establish the following variant of the minimax inequality \[ \max_{y\in B}\inf_{x\in A}f(y,x)\geq\inf_{x\in A}\sup_{y\in B_{0}}f(y,x), \] maxy is an element of Binfx is an element of Af(y,x)>= infx is an element of Asupy is an element of B0f(y,x), where, unlike the classical minimax theorem, the functions $ f(y,\cdot ) $ f(y,& sdot;) may not be convex for all $ y\in B $ y is an element of B but only for some of them (those in a possibly smaller set $ B_{0}), $ B0), while the function f is allowed to take infinite values. An application to the unique remoteness of sets and functions is given.
The purpose of this paper is to design a novel iterative algorithm to solve a generalized split feasibility and fixed point problem with multiple output sets (GSFFPPM) in the framework of Hilbert spaces. The proposed algorithm combines inertial extrapolation and the S-iterative methodology to accelerate convergence, Tikhonov regularization to ensure stability, and viscosity approximation to guarantee strong convergence to a solution of the GSFFPPM. Due to the generality of our model, we demonstrate the applicability of our iterative method to important classes of problems, including split variational inclusions and split equilibrium problems. To illustrate the practical relevance and computational efficiency of the proposed method, we present numerical experiments on real-world models such as Nash-Cournot semi-oligopolistic market equilibria and signal recovery tasks. These numerical experiments demonstrate the robustness and effectiveness of the proposed method.
This paper addresses sum-of-squares representations of nonnegative functions that are definable in o-minimal structures on $ (\mathbb {R}, +, \cdot ) $ (R,+,& sdot;). Namely, let \[ f, g_1, \ldots, g_l, h_1, \ldots, h_m \colon \mathbb{R}<^>n o \mathbb{R} \] f,g1,& mldr;,gl,h1,& mldr;,hm:Rn -> R be definable $ C<^>p $ Cp functions ( $ p \ge 2 $ p >= 2), and assume that f is nonnegative on the set \[ S := \{x \in \mathbb{R}<^>n \ | \ g_1(x) \ge 0, \ldots, g_l(x) \ge 0, h_1(x) = 0, \ldots, h_m(x) = 0 \}. \] S:={x is an element of Rn | g1(x)>= 0,& mldr;,gl(x)>= 0,h1(x)=0,& mldr;,hm(x)=0}. Under some natural hypotheses on zeros of f in S, we show that f is expressible in the form \[ f = \phi_0 + \sum_{i = 1}<^>l \phi_i g_i + \sum_{j =1}<^>m \psi_j h_j, \] f=phi 0+& sum;i=1l phi igi+& sum;j=1m psi jhj, where $ \phi _i, \psi _j \colon \mathbb {R}<^>n ightarrow \mathbb {R} $ phi i,psi j:Rn -> R are definable $ C<^>{p - 2} $ Cp-2-functions and each $ \phi _i $ phi i is a sum of squares of definable $ C<^>{p - 2} $ Cp-2-functions. As a consequence, we derive global optimality conditions which generalize the Karush-Kuhn-Tucker optimality conditions for nonlinear convex optimization.
We study the supportedness of nondominated points of multiobjective optimization problems, that is, whether they can be obtained via weighted sum scalarization. One key question is how supported points behave under an efficiency-preserving transformation of the original problem. Under a differentiability assumption, we characterize the transformations that preserve both efficiency and supportedness as the component-wise transformations with strictly increasing and convex components. In addition, we consider transformations that can render originally unsupported points supported in the transformed problem. This enables algorithms to find nondominated points by applying the weighted sum scalarization to a transformed problem.
In this paper, we introduce a new four-step iterative scheme, referred to as the Jungck-HR iteration, for approximating the unique common fixed point of a pair of contractive mappings in hyperbolic spaces. We establish strong convergence, stability, and Delta-convergence results for the proposed method. A comparative analysis shows that the Jungck-HR iteration converges faster than the Jungck-AI and Jungck-DK iterative schemes and remains convergent for certain contractive mappings where the Jungck-AI iteration fails. Numerical experiments are presented to demonstrate the convergence behaviour of the generated sequences. In addition, we conduct numerical simulations and present graphical illustrations showing the convergence of the orbits under the Jungck-HR iteration, thereby extending its theoretical applicability. As an application, the effectiveness of the proposed scheme is demonstrated by solving a two-dimensional nonlinear Volterra integral equation.
The filled function method is an effective approach to find the global optimal solution of global optimization problems by using the local search algorithm. The characteristics of the filled function significantly contribute to the efficiency of this kind of method. In this paper, we propose a novel filled function with one-parameter, which overcomes the deficiencies of some previous filled functions such as discontinuity and nondifferentiability, only containing local information of objective function and difficulty for the choice of the parameter. Then, a new filled function method for unconstrained global optimization is designed based on the properties of the introduced filled function. Furthermore, numerical experiments on a number of test problems are conducted to illustrate the efficiency and reliability of our algorithm. Finally, we extend the proposed algorithm to solve systems of nonlinear equations and satisfactory numerical performance are demonstrated.
Improved modulus-based matrix splitting iteration methods further advance the modulus-based matrix splitting iteration methods and the fast modulus-based matrix splitting iteration methods. Indeed, contrarily to other modulus-based strategies for implicit complementarity problems, such method do not require an inner-outer iteration, meaning that each iteration does not need the solution of a new linear system. In the paper, our methods with the parallel techniques are carried out to address the implicit complementarity problems, which further increases the efficiency. The convergence analysis and numerical experiments are given.
This paper presents an alternated inertial subgradient extragradient projection method for solving variational inequalities in real Hilbert spaces. The proposed algorithm employing an adaptive halfspace correction parameter at each iteration, has the advantage of improving stability and enhancing convergence. Under suitable conditions, weak convergence and strong convergence are established. Numerical experiments validate the effectiveness of the proposed method compared against existing related algorithms.
