Closed Convex Constraint Research Articles

A subgradient proximal method for solving a class of monotone multivalued variational inequality problems

It is well known that the algorithms with using a proximal operator can be not convergent for monotone variational inequality problems in the general case. Malitsky (Optim. Methods Softw. 33 (1) 140–164, ??) proposed a proximal extrapolated gradient algorithm ensuring convergence for the problems, where the constraints are a finite-dimensional vector space. Based on this proximal extrapolated gradient techniques, we propose a new subgradient proximal iteration method for solving monotone multivalued variational inequality problems with the closed convex constraint. At each iteration, two strongly convex subprograms are required to solve separately by using proximal operators. Then, the algorithm is convergent for monotone and Lipschitz continuous cost mapping. We also use the proposed algorithm to solve a jointly constrained Cournot-Nash equilibirum model. Some numerical experiment and comparison results for convex nonlinear programming confirm efficiency of the proposed modification.

Robust Approximate Optimality Conditions for Uncertain Nonsmooth Optimization with Infinite Number of Constraints

This paper deals with robust quasi approximate optimal solutions for a nonsmooth semi-infinite optimization problems with uncertainty data. By virtue of the epigraphs of the conjugates of the constraint functions, we first introduce a robust type closed convex constraint qualification. Then, by using the robust type closed convex constraint qualification and robust optimization technique, we obtain some necessary and sufficient optimality conditions for robust quasi approximate optimal solution and exact optimal solution of this nonsmooth uncertain semi-infinite optimization problem. Moreover, the obtained results in this paper are applied to a nonsmooth uncertain optimization problem with cone constraints.

A constrained state space approach for estimating firm efficiency

In this study a panel data state space approach is used to estimate efficiencies of productive units where the states are required to lie in a closed convex constraint set. The constraints are imposed by using oblique projections of state variables onto the constraint set. This approach provides a flexible approximation that can capture the time-varying pattern of firm specific efficiency.

Almost sure convergence of random projected proximal and subgradient algorithms for distributed nonsmooth convex optimization

Two distributed algorithms are described that enable all users connected over a network to cooperatively solve the problem of minimizing the sum of all users’ objective functions over the intersection of all users’ constraint sets, where each user has its own private nonsmooth convex objective function and closed convex constraint set, which is the intersection of a number of simple, closed convex sets. One algorithm enables each user to adjust its estimate using the proximity operator of its objective function and the metric projection onto one constraint set randomly selected from a number of simple, closed convex sets. The other determines each user’s estimate using the subdifferential of its objective function instead of the proximity operator. Investigation of the two algorithms’ convergence properties for a diminishing step-size rule revealed that, under certain assumptions, the sequences of all users generated by each of the two algorithms converge almost surely to the same solution. It also showed that the rate of convergence depends on the step size and that a smaller step size results in quicker convergence. The results of numerical evaluation using a nonsmooth convex optimization problem support the convergence analysis and demonstrate the effectiveness of the two algorithms.

Constrained consensus of asynchronous discrete-time multi-agent systems with time-varying topology

In this paper, the constrained consensus problem is studied for asynchronous discrete-time multi-agent system, where each agent needs to lie in a closed convex constraint set while reaching consensus. The communication graphs are assumed to be directed, unbalanced, dynamically changing. In addition, their union graph is assumed to be strongly connected among certain intervals of finite length. To deal with the asynchronous communications among agents, the original asynchronous system is equivalently transformed to the synchronous one by adding new agents. By employing the properties of the projection on the convex sets, the distance from the states of the agents in the newly constructed system to the intersection set of all agents’ constraint sets is estimated. Based on this estimation, the original system is proven to reach consensus by showing that the linear parts of the newly constructed systems converge and the nonlinear parts vanish over time. Finally, two numerical examples are provided to show the effectiveness of the theoretical results.

Strict Fejér Monotonicity by Superiorization of Feasibility-Seeking Projection Methods

We consider the superiorization methodology, which can be thought of as lying between feasibility-seeking and constrained minimization. It is not quite trying to solve the full-fledged constrained minimization problem; rather, the task is to find a feasible point which is superior (with respect to the objective function value) to one returned by a feasibility-seeking only algorithm. Our main result reveals new information about the mathematical behavior of the superiorization methodology. We deal with a constrained minimization problem with a feasible region, which is the intersection of finitely many closed convex constraint sets, and use the dynamic string-averaging projection method, with variable strings and variable weights, as a feasibility-seeking algorithm. We show that any sequence, generated by the superiorized version of a dynamic string-averaging projection algorithm, not only converges to a feasible point but, additionally, also either its limit point solves the constrained minimization problem or the sequence is strictly Fejér monotone with respect to a subset of the solution set of the original problem.

Constrained Consensus in Unbalanced Networks With Communication Delays

In this note, a constrained consensus problem is studied for multi-agent systems in unbalanced networks in the presence of communication delays. Here each agent needs to lie in a closed convex constraint set while reaching a consensus. The communication graphs are directed, dynamically changing, and not necessarily balanced and only the union of the graphs is assumed to be strongly connected among each time interval of a certain bounded length. The analysis is performed based on an undelayed equivalent system that is composed of a linear main body and an error auxiliary. To tackle the loss of symmetry caused by unbalanced graphs and communication delays, a novel approach is proposed. The idea is to estimate the distance from each agent to the intersection set of all agents' constraint sets based on the properties of the projection on convex sets so as to show consensus convergence by contradiction. It is shown that the error auxiliary vanishes as time evolves and the linear main body converges to a vector with an exponential rate as a separate system. It is also shown that the communication delays do not affect the consensus stability and constrained consensus is reached even if the communication delays are arbitrarily bounded. Finally, a numerical example is included to illustrate the obtained theoretical results.

Consensus-based distributed optimisation of multi-agent networks via a two level subgradient-proximal algorithm

This paper presents a consensus-based stochastic subgradient algorithm for multi-agent networks to minimise multiple convex but not necessarily differential objective functions, subject to an intersection set of multiple closed convex constraint sets. Compared with the existing results an alternative subgradient algorithm is first introduced based on two level subgradient iterations, where the first level is to minimise the component functions, and the second to enforce the iterates not oscillate from the constraint set wildly. In addition, a distributed consensus-based type of the proposed subgradient algorithm is constructed within the framework of multi-agent networks for the case when the iteration index of local objective functions and local constraint sets is not homologous. Detailed convergence analysis of the proposed algorithms is established using matrix theories and super-martingale convergence theorem. In addition, a pre-step convergence factor is obtained in this study to characterise the distance between the iterations and the optimal set, while some existing literatures only present a convergence work. Simulation results are given to demonstrate the effectiveness of the developed theoretical results.

Optimal investment-consumption problem with constraint

In this paper, we consider an optimal investment-consumption problem subject to a closed convex constraint. In the problem, a constraint is imposed on both the investment and the consumption strategy, rather than just on the investment. The existence of solution is established by using the Martingale technique and convex duality. In addition to investment, our technique embeds also the consumption into a family of fictitious markets. However, with the addition of consumption, it leads to nonreflexive dual spaces. This difficulty is overcome by employing the so-called technique of ``relaxation-projection' to establish the existence of solution to the problem. Furthermore, if the solution to the dual problem is obtained, then the solution to the primal problem can be found by using the characterization of the solution. An illustrative example is given with a dynamic risk constraint to demonstrate the method.

Optimal control of ODE systems involving a rate independent variational inequality

This paper is concerned with an optimal control problem for a systemof ordinary differential equations with rate independent hysteresismodelled as a rate independent evolution variational inequalitywith a closed convex constraint $Z\subset \mathbb{R}^m$.We prove existence of optimal solutions as well as necessary optimalityconditions of first order. In particular, under certain regularityassumptions we completely characterize the jump behaviour of the adjoint.

Nested Iterative Algorithms for Convex Constrained Image Recovery Problems

The objective of this paper is to develop methods for solving image recovery problems subject to constraints on the solution. More precisely, we will be interested in problems which can be formulated as the minimization over a closed convex constraint set of the sum of two convex functions f and g, where f may be nonsmooth and g is differentiable with a Lipschitz-continuous gradient. To reach this goal, we derive two types of algorithms that combine forward-backward and Douglas–Rachford iterations. The weak convergence of the proposed algorithms is proved. In the case when the Lipschitz-continuity property of the gradient of g is not satisfied, we also show that, under some assumptions, it remains possible to apply these methods to the considered optimization problem by making use of a quadratic extension technique. The effectiveness of the algorithms is demonstrated for two wavelet-based image restoration problems involving a signal-dependent Gaussian noise and a Poisson noise, respectively.

The application of an oblique-projected Landweber method to a model of supervised learning

This paper brings together a novel information representation model for use in signal processing and computer vision problems, with a particular algorithmic development of the Landweber iterative algorithm. The information representation model allows a representation of multiple values for a variable as well as an expression for confidence. Both properties are important for effective computation using multi-level models, where a choice between models will be implementable as part of the optimization process. It is shown that in this way the algorithm can deal with a class of high-dimensional, sparse, and constrained least-squares problems, which arise in various computer vision learning tasks, such as object recognition and object pose estimation. While the algorithm has been applied to the solution of such problems, it has so far been used heuristically. In this paper we describe the properties and some of the peculiarities of the channel representation and optimization, and put them on firm mathematical ground. We consider the optimization a convexly constrained weighted least-squares problem and propose for its solution a projected Landweber method which employs oblique projections onto the closed convex constraint set. We formulate the problem, present the algorithm and work out its convergence properties, including a rate-of-convergence result. The results are put in perspective with currently available projected Landweber methods. An application to supervised learning is described, and the method is evaluated in an experiment involving function approximation, as well as application to transient signals.

On Existence of Nash Equilibria of Games with Constraints on Multistrategies

In this paper, sufficient conditions are given, which are less restrictivethan those required by the Arrow–Debreu–Nash theorem, on theexistence of a Nash equilibrium of an n-player game {1, . . . , Yn,f1, . . . , fn} in normal form with a nonempty closedconvex constraint C on the set Y=Πi Yi of multistrategies. Theith player has to minimize the function fi with respect to the ithvariable. We consider two cases. In the first case, Y is a real Hilbert space and the loss function class isquadratic. In this case, the existence of a Nash equilibrium is guaranteedas a simple consequence of the projection theorem for Hilbert spaces. In thesecond case, Y is a Euclidean space, the loss functions are continuous, andfi is convex with respect to the ith variable. In this case, the techniqueis quite particular, because the constrained game is approximated with asequence of free games, each with a Nash equilibrium in an appropriatecompact space X. Since X is compact, there exists a subsequence of theseNash equilibrium points which is convergent in the norm. If thelimit point is in C and if the order of convergence is greater than one,then this is a Nash equilibrium of the constrained game.

Regularization of ill-posed constrained minimization problems by iteration methods

The ill-posed minimization problems in Hilbert space with quadratic objective function and closed convex constraint set are considered. For the compact set the regularization methods for such problems are well understood [1, 2] The regularizing properties of some Iteration projection methods for noncompact constraint set are the main issues of this paper. We are looking the gradient projection method for the sphere.

First and Second-Order Optimality Conditions for Convex Composite Multiobjective Optimization

Multiobjective optimization is a useful mathematical model in order to investigate real-world problems with conflicting objectives, arising from economics, engineering, and human decision making. In this paper, a convex composite multiobjective optimization problem, subject to a closed convex constraint set, is studied. New first-order optimality conditions for a weakly efficient solution of the convex composite multiobjective optimization problem are established via scalarization. These conditions are then extended to derive second-order optimality conditions.

A Dual Approach to Linear Inverse Problems with Convex Constraints

A simple constraint qualification is developed and used to derive an explicit solution to a constrained optimization problem in Hilbert space. A finite parameterization is obtained for the minimum norm element in the intersection of a linear variety of finite co-dimension and a closed convex constraint set. The result extends previous duality theorems for convex cone set constraints. A fixed point iteration is presented for computing the parameters and yields a least-squares solution when the variety and constraint set have empty intersection. Proofs rely on nearest-point projections onto convex sets and the properties of monotone, firmly nonexpansive, and averaged mappings.

Iteration methods for convexly constrained ill-posed problems in hilbert space

Minimization problems in Hilbert space with quadratic objective function and closed convex constraint set C are considered. In case the minimum is not unique we are looking for the solution of minimal norm. If a problem is ill-posed, i.e. if the solution does not depend continuously on the data, and if the data are subject to errors then it has to be solved by means of regularization methods. The regularizing properties of some gradient projection methods—i.e. convergence for exact data, order of convergence under additional assumptions on the solution and stability for perturbed data—are the main issues of this paper.

On the existence of solutions to a class of optimization problems

A sufficient condition for the existence and uniqueness of solutions to a class of optimization problems in nonlinear programming form, with strictly convex cost functions, convex inequality and linear equality side constraints, and closed convex constraint sets is studied.