Convexity Assumptions Research Articles

Some (p, q)-Estimates of Hermite-Hadamard-Type Inequalities for Coordinated Convex and Quasi- Convex Functions

In this paper, we present the preliminaries of ( p , q ) -calculus for functions of two variables. Furthermore, we prove some new Hermite-Hadamard integral-type inequalities for convex functions on coordinates over [ a , b ] × [ c , d ] by using the ( p , q ) -calculus of the functions of two variables. Furthermore, we establish an identity for the right-hand side of the Hermite-Hadamard-type inequalities on coordinates that is proven by using the ( p , q ) -calculus of the functions of two variables. Finally, we use the new identity to prove some trapezoidal-type inequalities with the assumptions of convexity and quasi-convexity on coordinates of the absolute values of the partial derivatives defined in the ( p , q ) -calculus of the functions of two variables.

On a Neumann problem for variational functionals of linear growth

We consider a Neumann problem for strictly convex variational functionals of linear growth. We establish the existence of minimisers among $\operatorname{W}^{1,1}$-functions provided that the domain under consideration is simply connected. Hence, in this situation, the relaxation of the functional to the space of functions of bounded variation, which has better compactness properties, is not necessary. Similar $\operatorname{W}^{1,1}$-regularity results for the corresponding Dirichlet problem are only known under rather restrictive convexity assumptions limiting its non-uniformity up to the borderline case of the minimal surface functional, whereas for the Neumann problem no such quantified version of strong convexity is required.

On a modified optimal control problem with first-order PDE constraints and the associated saddle-point optimality criterion

In this paper, based on a new multidimensional optimal control problem, in short (OCP), we introduce a modified multidimensional optimal control problem (which is simpler or easier to solve than the original one) involving first-order partial differential equations (PDEs) and inequality-type constraints. Optimality conditions associated with this new optimal control problem are formulated and proved, as well. Furthermore, under some generalized convexity assumptions, we establish an equivalence between an optimal solution of (OCP) and a saddle-point associated with the Lagrange functional corresponding to the modified multidimensional optimal control problem. Also, in order to illustrate the main results and their effectiveness, some applications are presented.

Translators asymptotic to cylinders

Abstract We show that the Bowl soliton in ℝ 3 {\mathbb{R}^{3}} is the unique translating solution of the mean curvature flow whose tangent flow at - ∞ {-\infty} is the shrinking cylinder. As an application, we show that for a generic mean curvature flow, all (non-static) translating limit flows are the bowl soliton. The crucial point is that we do not make any global convexity assumption, while as the same time, the asymptotic requirement is very weak.

Karush–Kuhn–Tucker optimality conditions and duality for convex semi-infinite programming with multiple interval-valued objective functions

This paper deals with convex semi-infinite programming with multiple interval-valued objective functions. We first investigate necessary and sufficient Karush–Kuhn–Tucker optimality conditions for some types of optimal solutions. Then, we formulate types of Mond–Weir and Wolfe dual problems and explore duality relations under convexity assumptions. Some examples are provided to illustrate the advantages of our results in some cases.

First- and second-order necessary conditions with respect to components for discrete optimal control problems

This paper is devoted to the study of discrete optimal control problems. We aim to obtain more constructive optimality conditions under weakened convexity assumptions. Based on a new approach introduced in this work, an optimality condition with respect to every component is obtained in the form of a global maximum principle. In addition, an optimality condition with respect to one of the components of a control in the form of the global maximum principle and with respect to another component of a control in the form of the linearized maximum principle are obtained. Furthermore, various second-order optimality conditions in terms of singular and quasi-singular controls with respect to the components are obtained on the fly.

Unique equilibrium in contests with incomplete information

Considered are imperfectly discriminating contests in which players may possess private information about the primitives of the game, such as the contest technology, valuations of the prize, cost functions, and budget constraints. We find general conditions under which a given contest of incomplete information admits a unique pure-strategy Nash equilibrium. In particular, provided that all players have positive budgets in all states of the world, existence requires only the usual concavity and convexity assumptions. Information structures that satisfy our conditions for uniqueness include independent private valuations, correlated private values, pure common values, and examples of interdependent valuations. The results allow dealing with inactive types, asymmetric equilibria, population uncertainty, and the possibility of resale. It is also shown that any player that is active with positive probability ends up with a positive net rent.

Convergence Analysis of the Generalized Splitting Methods for a Class of Nonconvex Optimization Problems

In this paper, we propose generalized splitting methods for solving a class of nonconvex optimization problems. The new methods are extended from the classic Douglas–Rachford and Peaceman–Rachford splitting methods. The range of the new step-sizes even can be enlarged two times for some special cases. The new methods can also be used to solve convex optimization problems. In particular, for convex problems, we propose more relax conditions on step-sizes and other parameters and prove the global convergence and iteration complexity without any additional assumptions. Under the strong convexity assumption on the objective function, the linear convergence rate can be derived easily.

A Coordinate Optimization Approach for Concurrent Design

In concurrent design (CCD), multiple design teams execute their tasks simultaneously and then exchange information to update their designs. The process then iterates until the termination criterion is met. When properly controlled and executed, CCD can be an effective method to shorten the time in product development for complex and large-scale projects thanks to its parallel nature. In this note, we propose a coordinate optimization framework to model and control team coordination through information sharing in CCD. It can be shown that under a certain convexity assumption, CCD converges to a globally optimal design if the information sharing intensity is smaller than a certain threshold. Numerical experiments substantiate the theoretical results.

Convergence Analysis of a Distributed Optimization Algorithm with a General Unbalanced Directed Communication Network

In this paper, we discuss a class of distributed constrained optimization problems in power systems where the target is to optimize the sum of all agents’ local convex objective functions over a general unbalanced directed communication network. Each local convex objective function is known exclusively to a single agent, and the agents’ variables are constrained to global coupling linear constraint and individual box constraints. To collaboratively solve the optimization problems, existing distributed methods mostly require the communication network to be balanced or have the knowledge of in-neighbors’ out-degree for all agents, which are quite restrictive and hardly inevitable in practical applications. In contrast, we investigate a novel distributed primal-dual augmented (sub)gradient algorithm which utilizes a row-stochastic matrix (does not need each agent to know its in-neighbors out-degree) and employs uncoordinated step-sizes, and yet exactly converges to the optimal solution over a general unbalanced directed communication network. Under the assumptions of the strong convexity and smoothness on the aggregate objective functions, it is proved that the algorithm geometrically converges to the optimal solution if the uncoordinated step-sizes do not exceed the upper bound. An explicit analysis for the convergence rate of the proposed algorithm is also characterized. To manifest effectiveness and applicability of the proposed algorithm, three case studies are presented to solve two practical problems in power systems.

A New Search Direction via Hybrid Conjugate Gradient Coefficient for Solving Nonlinear System of Equations

Conjugate gradient methods are widely used for unconstrained optimization problems. Most of the conjugate gradient methods do not always generate a descent search direction. In this article, a new search direction via hybrid conjugate gradient method by convex combination of the earlier works is used and extended for the solution of nonlinear system of equations. The proposed search direction generates a descent direction at every iteration and without convexity assumption on the objective function. The new method possesses, preserved and inherits the properties of the Conjugate Gradient (CG) coefficient used. The proposed hybridized method played a useful, vital and powerful role in solving large-scale nonlinear system of equations as indicated in the numerical results.

Persuading Risk-Conscious Agents: A Geometric Approach

We consider a persuasion problem between a sender and a non-expected utility maximizing receiver whose utility may be nonlinear in her belief; we call such receivers risk-conscious. Such utility models arise, for example, when the receiver exhibits sensitivity to the variance of the payoff on choosing an action (e.g., uncertainty-aversion when waiting for a service). Due to this nonlinearity, the revelation principle fails and action recommendations no longer suffice for optimal persuasion. To overcome this challenge, we develop an optimization framework using the underlying geometry of the persuasion problem to pose it as a convex optimization program. We use this approach to analyze the setting of binary persuasion, where the receiver has two actions and the sender prefers one of them over the other in every state. Under a mild convexity assumption, we reduce the persuasion problem to a linear program, and establish a canonical basis for the set of signals in an optimal signaling scheme. The signals in this canonical set either reveal the state, or induce in the receiver uncertainty between two states. Finally, we apply our methods to optimally share waiting time information in a queueing system with uncertainty-averse customers.

Monotonicity and conformality in multicommodity network‐flow problems

AbstractThe objective of this paper is to develop a monotonicity theory for the important class of minimum convex‐cost parametric multicommodity network‐flow problems defined over directed graphs. The results allow us to determine when it is possible to predict, without numerical computations, the direction of change of optimal multicommodity flows resulting from changes in arc‐commodity parameters. In particular, we provide necessary and sufficient conditions that for every cost function satisfying some convexity and submodularity assumptions there always exists an optimal multicommodity flow for which the flow of a commodity in a given arc a is nondecreasing (resp., nonincreasing) in the parameter of a distinct commodity in arc b. These conditions are that either (1) there are only two commodities and the underlying undirected graph is series‐parallel or (2) there are three or more commodities and the graph is 2‐isomorphic to a suspension graph. A characterization of the precise pairs of arcs for which the above monotonicity result holds is also provided.

-minimization methods for image restoration problems based on wavelet frames

In this paper we consider a class of -minimization and wavelet frame-based models for image deblurring and denoising. Mathematically, they can be formulated as minimizing the sum of a data fidelity term and the l0-‘norm’ of the framelet coefficients of the underlying image, and we are particularly interested in three different types of data fidelity forms for image restoration problems. We first study the first-order optimality conditions for these models. We then propose a penalty decomposition (PD) method for solving these problems in which a sequence of penalty subproblems are solved by a block coordinate descent (BCD) method. Under some suitable assumptions, we establish that any accumulation point of the sequence generated by the PD method satisfies the first-order optimality conditions of these problems. Moreover, for the problems in which the data fidelity term is convex, we show that such an accumulation point is a local minimizer of the problems. In addition, we show that any accumulation point of the sequence generated by the BCD method is a block coordinate minimizer of the penalty subproblem. Furthermore, under some convexity assumptions on the data fidelity term, we prove that such an accumulation point is a local minimizer of the penalty subproblem. Numerical simulations show that the proposed -minimization methods enjoy great potential for image deblurring and denoising in terms of solution quality and/or speed.

Extragradient and linesearch methods for solving split feasibility problems in Hilbert spaces

Utilizing the Tikhonov regularization method and extragradient and linesearch methods, some new extragradient and linesearch algorithms have been introduced in the framework of Hilbert spaces. In the presented algorithms, the convexity of optimization subproblems is assumed, which is weaker than the strong convexity assumption that is usually supposed in the literature, and also, the auxiliary equilibrium problem is not used. Some strong convergence theorems for the sequences generated by these algorithms have been proven. It has been shown that the limit point of the generated sequences is a common element of the solution set of an equilibrium problem and the solution set of a split feasibility problem in Hilbert spaces. To illustrate the usability of our results, some numerical examples are given. Optimization subproblems in these examples have been solved by FMINCON toolbox in MATLAB.

Quenched convergence and strong local equilibrium for asymmetric zero-range process with site disorder

We study asymmetric zero-range processes on Z with nearest-neighbour jumps and site disorder. The jump rate of particles is an arbitrary but bounded nondecreasing function of the number of particles. We prove quenched strong local equilibrium at subcritical and critical hydrodynamic densities, and dynamic local loss of mass at supercritical hydrodynamic densities. Our results do not assume starting from local Gibbs states. As byproducts of these results, we prove convergence of the process from given initial configurations with an asymptotic density of particles to the left of the origin. In particular , we relax the weak convexity assumption of [7, 8] for the escape of mass property. 1 MSC 2010 subject classification: 60K35, 82C22.

On the homogenization of means

The aim of this paper is to introduce several notions of homogenization in various classes of weighted means, which include quasiarithmetic and semideviation means. In general, the homogenization is an operator which attaches a homogeneous mean to a given one. Our results show that, under some regularity or convexity assumptions, the homogenization of quasiarithmetic means are power means, and homogenization of semideviation means are homogeneous semideviation means. In other results, we characterize the comparison inequality, the Jensen concavity, and Minkowski- and H\"older-type inequalities related to semideviation means.

A modified descent Polak–Ribiére–Polyak conjugate gradient method with global convergence property for nonconvex functions

Following the modification scheme of Dong et al. made on the Hestenes–Stiefel method, we suggest a modified Polak–Ribiere–Polyak technique which satisfies the sufficient descent condition. We show that the method is globally convergent with the Wolfe line search conditions as well as the backtracking Armijo-type line search strategy proposed by Grippo and Lucidi, without convexity assumption on the objective function. Numerical experiments on some test functions of the CUTEr collection show that the method performs promisingly.

Distributed Strategy for Optimal Dispatch of Unbalanced Three-Phase Islanded Microgrids

This paper presents a distributed strategy for the optimal dispatch of islanded microgrids, modeled as unbalanced three-phase electrical distribution systems. To set the dispatch of the distributed generation (DG) units, an optimal generation problem is stated and solved distributively based on primal–dual constrained decomposition and a first-order consensus protocol, where units can communicate only with their neighbors. Thus, convergence is guaranteed under the common convexity assumptions. The islanded microgrid operates with the standard hierarchical control scheme, where two control modes are considered for the DG units: a voltage control mode, with an active droop control loop, and a power control mode, which allows setting the output power in advance. To assess the effectiveness and flexibility of the proposed approach, simulations were performed in a 25-bus unbalanced three-phase microgrid. According to the obtained results, the proposed strategy achieves a lower cost solution when compared with a centralized approach based on a static droop framework, with a considerable reduction on the communication system complexity. Additionally, it corrects the mismatch between generation and consumption even during the execution of the optimization process, responding to changes in the load consumption, renewable generation, and unexpected faults in units.

SSB representation of preferences: Weakening of convexity assumptions

A continuous skew-symmetric bilinear (SSB) representation of preferences has recently been proposed in a topological vector space, assuming a weaker notion of convexity of preferences than in the classical (algebraic) case. Equipping a linear vector space with the so-called inductive linear topology, we derive the algebraic SSB representation on such topological basis, thus weakening the convexity assumption. Such a unifying approach to SSB representation leads, moreover, to a stronger existence result for a maximal element and opens a way for a non-probabilistic interpretation of the algebraic theory. Note finally that our method of using powerful topological techniques to derive purely algebraic result may be of general interest.