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.
The tractability of optimization problems depends critically on structural properties of the objective function. Convexity guarantees global optimality of local solutions and enables polynomial-time algorithms under mild assumptions, but many problems arising in modern applications-particularly in machine learning-are inherently nonconvex. Remarkably, a large class of such problems remains amenable to efficient optimization due to additional structure that weakens or generalizes convexity without forfeiting favorable algorithmic behaviour. This paper surveys and systematizes notions of convexity and its generalizations, while also providing new comparative insights and explicit inclusion relationships among these function classes. We present a coherent taxonomy of functions that generalize, strengthen, and relax convexity, consolidating definitions, equivalent characterizations, closure properties, and hierarchical relations that are currently scattered across the optimization, operations research, and machine learning literature. Particular emphasis is placed on quasar-convexity, a recently introduced geometric condition that captures structured nonconvexity while enabling convergence guarantees comparable to those of convex optimization for many first-order methods. Through explicit inclusion diagrams and systematic comparisons, we clarify the relationships among classical generalizations, geometric variants, regularity conditions, and partial convexity notions. The resulting 'Convexity Zoo' provides a comprehensive reference for researchers seeking to understand and exploit structured nonconvexity in contemporary optimization.
In the context of Banach spaces, we present a new and more general class of generalized enriched Kannan-type cyclic contraction mapping and prove relevant fixed point theorems. Our approach is distinct in that we use the Krasnoselskij iteration to approximate the fixed point, which allows us to generate explicit error estimates and convergence behaviours. We build generalized iterated function systems (IFS) produced by these enriched contractions and demonstrate the existence and uniqueness of their attractors as a novel application. The present study is unique in that it links classical contraction concepts to the theory of IFS and extends them to a more complex cyclic system. In addition to a numerical illustration, examples are provided to demonstrate the efficacy of our findings.
This paper proposes a novel variant of the proximal gradient method for a constrained multiobjective composite optimization problem, in which each objective function is the sum of a smooth function and a non-differentiable one. At each iteration, the descent direction of the differentiable part is calculated by solving a quadratic subproblem, and the non-differentiable part draws on the idea of the multi-objective proximal point method. Under locally Lipschitz continuity assumption, it is proved that the accumulation points of the sequence generated by the algorithm are Pareto critical. Under convexity condition, it is further proved that the sequence generated by the algorithm converges to a weak Pareto efficient point. We also establish the global convergence rates of the proposed approach. More specifically, we present the global convergence rates of $ \mathcal {O}(\frac {1}{k}) $ O(1k) for convex case and $ \mathcal {O}(r<^>{k}) $ O(rk) with some $ r\in (0,1) $ r is an element of(0,1) for strongly convex case, respectively. In addition, an application to a binary classification problem in supervised machine learning is given to validate the efficiency of the proposed method. Finally, performance experiments suggest that the proposed algorithm can robustly generate Pareto fronts of multiple synthesis test problems compared with existing ones.
This paper investigates a Bregman projection algorithm for solving split variational inequality problems with multiple output sets in real Hilbert spaces, where the underlying operators are assumed to be pseudomonotone and not necessarily Lipschitz continuous. This relaxation considerably broadens the applicability of the proposed method. The algorithm, inspired by the Halpern iteration, the CQ algorithm, and Tseng's extragradient technique, incorporates a two-step inertial strategy to accelerate convergence. A strong convergence theorem is established without requiring prior knowledge of the operator norms. Numerical experiments, together with graphical illustrations, are provided to demonstrate the efficiency of the proposed algorithm in comparison with existing methods. In particular, an application to a signal recovery problem is included to highlight the practical relevance of the approach.
We consider a history-dependent hemivariational inequality in a reflexive Banach space X, stated on the interval of time [0,T] with T>0 and governed by a time-dependent set of constraints. We use arguments of pseudomonotonicity, Mosco convergence and fixed point in order to provide the existence of a unique solution u is an element of C([0,T];X) of the inequality, together with a pointwise convergence result. Next, under additional assumptions, we provide necessary and sufficient conditions which guarantee the uniform convergence of a sequence of functions {u(n)} subset of C([0,T];X) to the solution u. We then introduce and study two well-posedness concepts for the corresponding inequality. Our results give rise to various applications. To provide an example, we illustrate their use in the study of a mathematical model which describes the equilibrium of a viscoelastic rod in contact with a rigid-deformable obstacle, the so-called foundation.
This paper proposes a new fixed point algorithm that combines a viscosity-type method with two-step inertial extrapolation to approximate common fixed points of a family of strongly quasi-nonexpansive mappings in real Hilbert spaces. We establish strong convergence under relaxed parameter conditions, without relying on-line rule of the inertial parameters. The proposed algorithm is applied to various classes of optimization problems, including variational inequalities, quasi-inclusion problems, convex minimization, and split feasibility problems. Finally, several numerical experiments, such as data classification on a heart disease dataset and comparisons with well-known machine learning algorithms are conducted to demonstrate the effectiveness and convergence behaviour of the method.
Robust optimization is one of the most important approaches for handling data uncertainty with minimal information in mathematical programming. In recent years, Pareto robust optimality has been developed as a promising alternative to the classical worst-case approach. One of the research gaps in this area is the limited study of problems with unbounded uncertainty sets. In order to address this gap, we study (Pareto) robust solutions of nonlinear programming problems with unbounded uncertainty sets. By invoking the Minkowski-Weyl theorem, we construct a compactification of the considered uncertainty set and establish useful connections between the robust and Pareto robust solutions under the approximated compact uncertainty set and the original unbounded one. An interesting contribution of this study is the construction of the compactification without the need to explicitly calculate the extreme points of the set. Furthermore, we formulate a deterministic multi-objective programming problem whose (weakly) efficient solutions are strongly connected with the (Pareto) robustly optimal solutions of the original uncertain problem. These results lead to an interactive algorithm that provides an approximation of the set of (Pareto) robust solutions. In addition to establishing several theoretical results and designing the algorithm, we illustrate our approach through numerical examples and an energy optimization problem. Our findings provide a tool to investigate whether an optimal solution found under stable market conditions, remains Pareto robustly optimal under market instability. In other words, we aim to determine whether such a solution remains reliable even if the uncertainty set significantly expands due to market shocks.
In this paper, we introduce an Anderson-accelerated method with a self-adaptive step size to solve variational inequality problem with a pseudo-monotone operator. The sequence generated by the proposed algorithm converges weakly to a solution of the variational inequality problem under some suitable conditions. Finally, some numerical experiments are performed to show the effectiveness of the method.
We propose two Tseng methods with double inertial steps and self-adaptive step sizes to approximate the zero of the sum of two maximal monotone operators in real Hilbert spaces. These methods are based on the inertial technique and regularization ideas. Unlike several existing approaches, our proposed methods do not require the Lipschitz continuity assumptions for the associated single-valued operator. Additionally, we employ a more efficient self-adaptive step size technique that generates non-monotonic sequence of step sizes, eliminating the need for time-consuming line search procedures. Under relaxed assumptions, we establish the strong convergence of the proposed algorithms and apply them to solve other optimization problems. Finally, numerical experiments demonstrate the comparative advantages of our methods over existing methods in the literature.
