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In this paper, we consider finite sum composite optimization problems with many functional constraints. The objective function is expressed as a finite sum of two terms, one of which admits easy computation of (sub)gradients while the other is amenable to proximal evaluations. We assume a generalized bounded gradient condition on the objective which allows us to simultaneously tackle both smooth and nonsmooth problems. We also consider the cases of both with and without a quadratic functional growth property. Further, we assume that each constraint set is given as the level set of a convex but not necessarily a differentiable function. We reformulate the constrained finite sum problem into a stochastic optimization problem for which the stochastic subgradient projection method from Necoara and Singh [Stochastic subgradient projection methods for composite optimization with functional constraints; 2022 Journal of Machine Learning Research, 23, 1–35] specializes in a collection of mini-batch variants, with different mini-batch sizes for the objective function and functional constraints, respectively. More specifically, at each iteration, our algorithm takes a mini-batch stochastic proximal subgradient step aimed at minimizing the objective function and then a subsequent mini-batch subgradient projection step minimizing the feasibility violation. By specializing in different mini-batching strategies, we derive exact expressions for the stepsizes as a function of the mini-batch size and in some cases we also derive insightful stepsize-switching rules which describe when one should switch from a constant to a decreasing stepsize regime. We also prove sublinear convergence rates for the mini-batch subgradient projection algorithm which depend explicitly on the mini-batch sizes and on the properties of the objective function. Numerical results also show a better performance of our mini-batch scheme over its single-batch counterpart. [ABSTRACT FROM AUTHOR]

In the smooth constrained optimization setting, this work introduces the Domain Complementary Approximate Karush–Kuhn–Tucker (DCAKKT) condition, inspired by a sequential optimality condition recently devised for non-smooth constrained optimization problems. It is shown that the augmented Lagrangian method can generate limit points satisfying DCAKKT, and it is proved that such a condition is not related to previously established sequential optimality conditions. An essential characteristic of the DCAKKT is to capture the asymptotic potential increasing of the Lagrange multipliers using a single parameter. Besides that, DCAKKT points satisfy the Strong Approximate Gradient Projection (SAGP) condition. Due to the intrinsic features of DCAKKT, which combine strength and generality, this novel and genuine sequential optimality condition may shed some light upon the practical performance of algorithms that are yet to be devised. [ABSTRACT FROM AUTHOR]

In this two-part study, we develop a general theory of the so-called exact augmented Lagrangians for constrained optimization problems in Hilbert spaces. In contrast to traditional nonsmooth exact penalty functions, these augmented Lagrangians are continuously differentiable for smooth problems and do not suffer from the Maratos effect, which makes them especially appealing for applications in numerical optimization. Our aim is to present a detailed study of various theoretical properties of exact augmented Lagrangians and discuss several applications of these functions to constrained variational problems, problems with PDE constraints, and optimal control problems. The first paper is devoted to a theoretical analysis of an exact augmented Lagrangian for optimization problems in Hilbert spaces. We obtain several useful estimates of this augmented Lagrangian and its gradient, and present several types of sufficient conditions for KKT-points of a constrained problem corresponding to locally/globally optimal solutions to be local/global minimizers of the exact augmented Lagrangian. [ABSTRACT FROM AUTHOR]

Based on a coupling conjugation scheme and the perturbational approach, we build Fenchel–Lagrange dual problem of a composed optimization model with infinite constraints in separated locally convex spaces. This paper has mainly two targets. One is to establish strong duality under a new regularity condition (RCA) and an extension closed-type condition (ECRCA). The e-convex counterpart of Fenchel–Moreau theorem plays a key role in analysing the relation between them. The other aim is to achieve the sufficient and necessary characterizations for total duality in terms of c-subdifferentials. For this purpose, a formula for ε-c-subdifferentials of a proper function composed with a linear continuous operator is proved and applied. [ABSTRACT FROM AUTHOR]

Optimization problems with least constraint violation have many important practical backgrounds. The related works in the literature normally focus on the norm square measure function and the classical augmented Lagrangian method is proved to be suitable for finding an approximate solution. This paper considers the constrained convex optimization problem with the least constraint violation under a general measure function. The properties of the conjugate dual associated with the measure function of the shifted problem are discussed through the relations between the dual function and the optimal value function. The differentiability of the dual function associated with the measure function is proved. The properties of augmented Lagrangian method induced by the measure function are characterized in terms of the dual function. The optimality conditions for the problem with the least constraint violation are established in term of the augmented Lagrangian. It is shown that the augmented Lagrangian method has the properties that the sequence of shift measure values is decreasing, the sequence of multipliers is unbounded, and the sequence of shifts converges to the least violated shift under an extra assumption. [ABSTRACT FROM AUTHOR]

This paper deals with a variational inequality problem over a solution set of another variational inequality problem, which essentially is called the double-hierarchical constrained optimization problem. Inertial-based algorithms are devised for this problem and solution theorems of strong convergence are established. Simulation results on numerical examples are given to illustrate the behaviour of our proposed algorithms compared with some existing ones. [ABSTRACT FROM AUTHOR]

A feasible rounding approach is a novel technique to compute good feasible points for mixed-integer optimization problems. The central idea of this approach is the construction of a continuously described inner parallel set for which any rounding of any of its elements is feasible in the original mixed-integer problem. It is known that this approach is promising for problems in which no equality constraints on integer variables appear. Yet, so far the potential of incorporating equality constraints with integer variables into this approach remained unclear. In this article, we close this gap by developing a reduction scheme that enables the application of feasible rounding approaches to problems in which such equality constraints occur. Our computational study on a large test bed of MIPLIB instances shows that this reduction is applicable to a relevant share of practical problems. Moreover, our results illustrate that a non-empty inner parallel set is possible, but less likely to occur for practical problems with equality constraints on integer variables. Finally, our results indicate that the application of a feasible rounding approach can be beneficial for the computation of good feasible points even under the occurrence of equality constraints on integer variables. [ABSTRACT FROM AUTHOR]

Asymptotic stationarity and regularity conditions turned out to be quite useful to study the qualitative properties of numerical solution methods for standard nonlinear and complementarity constrained programs. In this paper, we first extend these notions to nonlinear optimization problems with nonsmooth but Lipschitzian data functions in order to find reasonable notions of asymptotic stationarity and regularity in terms of Clarke's and Mordukhovich's subdifferential construction. Particularly, we compare the associated novel asymptotic constraint qualifications with already existing ones. The second part of the paper presents two applications of the obtained theory. On the one hand, we specify our findings for complementarity constrained optimization problems and recover recent results from the literature which demonstrates the power of the approach. Furthermore, we hint at potential extensions to or- and vanishing constrained optimization. On the other hand, we demonstrate the usefulness of asymptotic regularity in the context of bilevel optimization. More precisely, we justify a well-known stationarity system for affinely constrained bilevel optimization problems in a novel way. Afterwards, we suggest a solution algorithm for this class of bilevel optimization problems which combines a penalty method with ideas from DC-programming. After a brief convergence analysis, we present the results of some numerical experiments. [ABSTRACT FROM AUTHOR]

We propose a decentralized penalty method for general convex constrained multi-agent optimization problems. Each auxiliary penalized problem is solved approximately with a special parallel descent splitting method. The method can be implemented in a computational network where each agent sends information only to the nearest neighbours. Convergence of the method is established under rather weak assumptions. We also describe a specialization of the proposed approach to the feasibility problem. [ABSTRACT FROM AUTHOR]

We propose a new duality scheme based on a sequence of smooth minorants of the weighted- ℓ 1 penalty function, interpreted as a parametrized sequence of augmented Lagrangians, to solve non-convex constrained optimization problems. For the induced sequence of dual problems, we establish strong asymptotic duality properties. Namely, we show that (i) the sequence of dual problems is convex and (ii) the dual values monotonically increase to the optimal primal value. We use these properties to devise a subgradient based primal–dual method, and show that the generated primal sequence accumulates at a solution of the original problem. We illustrate the performance of the new method with three different types of test problems: A polynomial non-convex problem, large-scale instances of the celebrated kissing number problem, and the Markov–Dubins problem. Our numerical experiments demonstrate that, when compared with the traditional implementation of a well-known smooth solver, our new method (using the same well-known solver in its subproblem) can find better quality solutions, i.e. 'deeper' local minima, or solutions closer to the global minimum. Moreover, our method seems to be more time efficient, especially when the problem has a large number of constraints. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new inexact version of the projected subgradient method to solve nondifferentiable constrained convex optimization problems. The method combine ϵ-subgradient method with a procedure to obtain a feasible inexact projection onto the constraint set. Asymptotic convergence results and iteration-complexity bounds for the sequence generated by the method employing the well-known exogenous stepsizes, Polyak's stepsizes, and dynamic stepsizes are established. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of minimizing a continuously differentiable function subject to sparsity constraints. We formulate this problem as an equivalent disjunctive constrained optimization program. Then, we extend some of the well-known constraint qualifications by using the contingent and normal cones of the sparsity set and show that these constraint qualifications can be applied to obtain the first-order optimality conditions. In addition, we give the first-order sufficient optimality conditions by defining a new generalized convexity notion. Furthermore, we present the second-order necessary and sufficient optimality conditions for sparsity constrained optimization problems. Finally, we provide some examples and special cases to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

Semi-infinite programming can be used to model a large variety of complex optimization problems. The simple description of such problems comes at a price: semi-infinite problems are often harder to solve than finite nonlinear problems. In this paper, we combine a classical adaptive discretization method developed by Blankenship and Falk [Infinitely constrained optimization problems. J Opt Theory Appl. 1976;19(2):261–281. https://doi.org/10.1007/BF00934096] and techniques regarding a semi-infinite optimization problem as a bi-level optimization problem. We develop a new adaptive discretization method which combines the advantages of both techniques and exhibits a quadratic rate of convergence. We further show that a limit of the iterates is a stationary point, if the iterates are stationary points of the approximate problems. [ABSTRACT FROM AUTHOR]

In this paper, we consider the weak approximate solutions to constrained vector optimization problems with variable ordering structures. In terms of abstract subdifferentials, normal cones and coderivatives, we establish Lagrange rules for this kind of solutions. [ABSTRACT FROM AUTHOR]

In this article, a class of lower-order smoothed objective penalty functions is introduced to find locally optimal points for constrained optimization problems. The exactness of the new penalty functions is studied. Based on the current locally optimal points, a new class of penalty functions based on filling properties is proposed. This new penalty function can be used to find a better locally optimal point. The exactness and filling properties of this penalty function are proved in this paper. To do this, two algorithms are presented to find the locally and globally optimal points. Additionally, their convergence is proved under some mild conditions. Finally, numerical results are included to illustrate the applicability of the local and global optimization algorithms. [ABSTRACT FROM AUTHOR]

In this paper, an adaptation of the sequential quadratically constrained quadratic programming method is proposed to solve inequality constrained minimization. At each iteration, it solves a norm-relaxed quadratically constrained quadratic programming subproblem that uses an active set identification technique to reduce the scale and computational cost. By taking a valid line search, the iterates always get into the feasible set after a finite number of iterations, and followed by a suitable update rule for the penalty parameters. Under suitable conditions without the strict complementarity, our method has the global and superlinear convergence properties. In addition, numerical results are reported to demonstrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

The two-cardinality sparse constrained optimization problems include sparse optimization problems and constrained sparse optimization problems in many fields, such as signal processing, image processing, securities investment etc. A smoothing penalty function method and a smoothing objective penalty function method are studied for two-cardinality sparse constrained optimization problems respectively. Some error estimations are proved for the smoothing penalty function and the smoothing objective penalty function. Based on the smoothing penalty function and smoothing objective penalty function, two algorithms are designed to solve two-cardinality sparse constrained optimization problems and their convergence is proved respectively. Numerical results show that the two algorithms have the similar effectiveness in finding out an approximate solution for two-cardinality sparse constrained optimization problems. [ABSTRACT FROM AUTHOR]

We analyse the convergence of the gradient projection algorithm, which is finalized with the Newton method, to a stationary point for the problem of nonconvex constrained optimization min x ∈ S f (x) with a proximally smooth set S = { x ∈ R n : g (x) = 0 } , g : R n → R m and a smooth function f. We propose new Error bound (EB) conditions for the gradient projection method which lead to the convergence domain of the Newton method. We prove that these EB conditions are typical for a wide class of optimization problems. It is possible to reach high convergence rate of the algorithm by switching to the Newton method. [ABSTRACT FROM AUTHOR]

Their work is an extension of algorithms devised for equilibrium problems, proposed by Moudafi, and by Iusem and Sosa, to the more general context of quasi-equilibrium problems. We are very proud to present the special issue of Optimization related to the XIII Brazilian Workshop on Continuous Optimization, held in September 2019, celebrating Alfredo Iusem's 70th anniversary. Pedro Jorge S. Santos, Paulo Sérgio M. Santos, and Susana Scheimberg contributed their paper I A Newton-type method for Quasi-Equilibrium Problems and applications i , where they extend a Newton-type method for particular Equilibrium Problems to the setting of Quasi-Equilibrium Problems. [Extracted from the article]

This paper deals with the Frank–Wolfe algorithm to solve a special class of non-compact constrained optimization problems. The notion of asymptotic cone is one the main concept used to introduce the class of problems considered as well as to establish the well definition of the algorithm. This class of optimization problems, with closed and convex constraint set, are characterized by two conditions on the gradient of the objective function. The first one establishes that the gradient of the objective function is Lipschitz continuous, which is quite usual in the analysis of this algorithm. The second one, which is new in this subject, establishes that the gradient belongs to the interior of dual asymptotic cone of the constraint set. Classical results on asymptotic behaviour and iteration complexity bounds for the sequence generated by Frank–Wolfe algorithm are extended to this new class of problems. Some examples of problems with non-compact constraints and objective functions satisfying the aforementioned conditions are provided. [ABSTRACT FROM AUTHOR]

This paper is concerned with the inexact variable metric method for solving convex-constrained optimization problems. At each iteration of this method, the search direction is obtained by inexactly minimizing a strictly convex quadratic function over the closed convex feasible set. Here, we propose a new inexactness criterion for the search direction subproblems. Under mild assumptions, we prove that any accumulation point of the sequence generated by the new method is a stationary point of the problem under consideration. In order to illustrate the practical advantages of the new approach, we report some numerical experiments. In particular, we present an application where our concept of the inexact solutions is quite appealing. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the recession cone and recession function in the nonconvex case. By virtue of the recession function, we investigate the monotonicity of a function. Then, we obtain a characterization of the Lipschitz continuity for a function by using the recession function. As applications, we study the monotonicity and Lipschitz continuity properties of the optimal value function for a parametric constrained optimization problem. Moreover, we present a characterization of the nonemptiness and boundedness of the solution set for a constrained optimization problem without convexity assumption. [ABSTRACT FROM AUTHOR]

In this paper, we propose an algorithm for solving linearly constrained non-smooth, non-convex optimization problems. The objective functions in these problems are, in general, upper semidifferentiable locally Lipschitz functions. The method is based on the idea of adapting, to the non-smooth setting, the variant of the reduced gradient algorithm proposed by Luenberger, and on bundle techniques which are aimed at building an approximation of the subdifferential. It may be thought of as an extension of reduced gradient methods for dealing with both non-smoothness and non-convexity of the objective function. Under the non-degeneracy assumption, the termination of the proposed algorithm at a stationary point is proved. Numerical results and comparisons with some existing methods are reported to show the efficiency of our algorithm. [ABSTRACT FROM AUTHOR]

This paper mainly focuses on optimality conditions for efficient solutions of a nonconvex constrained multiobjective optimization problem via image space analysis. By virtue of the Gerstewitz function and the sum of the components of the objective function, a nonlinear regular weak separation function for efficient solutions, is constructed. In order to illustrate the importance of the sum of the components of the objective function in the nonlinear regular weak separation function for efficient solutions, a nonlinear regular weak separation function for weak efficient solutions, is also given. Moreover, a nonlinear strong separation function for efficient and weak efficient solutions, is introduced. Then, some theorems of the weak and strong alternative and a global necessary and sufficient optimality condition for efficient solutions are derived by means of the nonlinear regular weak and strong separation functions for efficient solutions. A saddle point sufficient optimality condition for efficient solutions in terms of a generalized Lagrangian function associated with the nonlinear regular weak separation function, is established. Finally, under some suitable assumptions, a saddle point necessary optimality condition is obtained, which further yields a Karush/Kuhn–Tucker necessary optimality condition for efficient solutions by the Clarke subdifferential. [ABSTRACT FROM AUTHOR]

The Barzilai–Borwein (BB) step size initially proposed in the context of unconstrained optimization has become one of the most popular step choices for gradient-based methods in the optimization community. It is well-known that the powerful variational inequality can be used to characterize the first-order optimality condition of constrained optimization problems. However, a variational inequality problem is not always equivalent to a constrained optimization problem. In this paper, we follow the spirit of BB step size and propose a projection method with alternate BB step size for general variational inequalities. Although the global convergence is established under some strong conditions, a series of computational experiments on nonlinear complementarity problems, image deblurring problems and generalized Nash equilibrium problems demonstrate that the proposed almost-parameter-free projection method is more efficient than some existing state-of-the-art projection methods in the literature. [ABSTRACT FROM AUTHOR]

This article is devoted to the investigation of a vector equilibrium problem involving equality, inequality and set constraints with nonsmooth functions via the higher-order Studniarski derivatives. Under the suitable constraint qualifications, higher-order Karush-Kuhn-Tucker type necessary efficiency conditions for the local weak efficient solutions of a constrained vector equilibrium problem are given. Under suitable assumptions on the objective and constraint functions, the higher-order Karush-Kuhn-Tucker type necessary optimality conditions for local weak efficient solutions become the sufficient conditions. As applications, we obtain the first-order and higher-order optimality conditions for the local weak efficient solutions of a constrained vector variational inequality problem, a constrained vector optimization problem and a transportation problem with two-sided constraints on supplies or demands in terms of Studniarski's derivatives. Several examples are also provided to illustrate the results of the paper. [ABSTRACT FROM AUTHOR]

Hemivariational inequalities arise in nonsmooth mechanics of solid involving nonmonotone and multi-valued mechanical relations. Typically, after the finite-element discretization, they lead to constrained nonsmooth nonconvex optimization problems with objective functions being the sum of quadratic functions and nonsmooth terms. In this paper, smoothing approximations are employed to solve the constrained nonsmooth nonconvex optimization problems. After properties of the smoothing functions are analysed, a smoothing quadratic regularization algorithm is presented and studied. The proposed algorithm can be implemented efficiently since the closed form solution is available at each iteration. Convergence of the algorithm is shown, and the worst-case complexity is investigated for reaching an ε-Clarke stationary point. A numerical example is reported to show the performance of the proposed method. [ABSTRACT FROM AUTHOR]

We propose a derivative-free algorithm for solving linear equality constrained non-linear optimization problems, named LECOA. In each iteration of the algorithm, the objective function is approximated by a quadratic model constructed from interpolation points. The choice of the points leaves some degree of freedom in the model taken up by minimizing the Frobenius norm of the change to the Hessian matrix of the model. The new iterate is generally generated by minimizing the model in a trust-region using a null space truncated conjugate gradient method. Numerical results are presented which show that the proposed algorithm competes against some algorithms in the literature. Experiments show that starting with the point that minimizes the infinity norm subject to the linear equality constraints gives excellent results. A limit-type global convergence of the proposed algorithm is proved under some reasonable assumptions. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyse a path-based incremental target level algorithm for minimizing a constrained convex optimization on complete Riemannian manifolds with lower bounded sectional curvature, where the object function consists of the sum of a large number of component functions. This algorithm extends, to the context of Riemannian manifolds, an incremental subgradient method emplying a version of dynamic stepsize rule. Some convergence results and iteration-complexity bounds of the algorithm are established. [ABSTRACT FROM AUTHOR]

An inner–outer approximation approach was recently developed to solve single chance constrained optimization (SCCOPT) problems. In this paper, we extend this approach to address joint chance constrained optimization (JCCOPT) problems. Using an inner–outer approximation, two smooth parametric optimization problems are defined whose feasible sets converge to the feasible set of JCCOPT from inside and outside, respectively. Any optimal solution of the inner approximation problem is a priori feasible to the JCCOPT. As the approximation parameter tends to zero, a subsequence of the solutions of the inner and outer problems, respectively, converge asymptotically to an optimal solution of the JCCOPT. As a main result, the continuous differentiability of the probability function of a joint chance constraint is obtained by examining the uniform convergence of the gradients of the parametric approximations. [ABSTRACT FROM AUTHOR]

We present an interior proximal method for solving constrained nonconvex optimization problems where the objective function is given by the difference of two convex function (DC function). To this end, we consider a linearized proximal method with a proximal distance as regularization. Convergence analysis of particular choices of the proximal distance as second-order homogeneous proximal distances and Bregman distances are considered. Finally, some academic numerical results are presented for a constrained DC problem and generalized Fermat–Weber location problems. [ABSTRACT FROM AUTHOR]

The scaled gradient projection (SGP) method, which can be viewed as a promising improvement of the classical gradient projection method, is a quite efficient solver for real-world problems arising in image science and machine learning. Most recently, Bonettini and Prato [Inverse Probl. 2015;31:095008. 20 p] proved that the SGP method with the monotone Armijo line search technique has the convergence rate, where k counts the iteration. In this paper, we first show that the SGP method could be equipped with the nonmonotone line search procedure proposed by Zhang and Hager [SIAM J Optim. 2004;14:1043-1056]. To some extent, such a nonmonotone technique might improve the performance of SGP method, because its effectiveness has been verified for unconstrained optimization by comparing with the traditional monotone and nonmonotone strategies. Then, we prove that the new SGP method also has the convergence rate under the condition that the objective function is convex. Furthermore, we derive the linear convergence of the SGP algorithm under the strongly convexity assumption of the involved objective function. [ABSTRACT FROM AUTHOR]

In this paper, based on a new smoothing function, the general box constrained variational inequalities are solved by a smoothing Newton algorithm with a nonmonotone line search. The proposed algorithm is proved to be globally and locally superlinearly convergent under suitable assumptions. The preliminary numerical results are reported. [ABSTRACT FROM AUTHOR]

We discuss several optimization procedures to solve finite element approximations of linear-quadratic Dirichlet optimal control problems governed by an elliptic partial differential equation posed on a 2D or 3D Lipschitz domain. The control is discretized explicitly using continuous piecewise linear approximations. Unconstrained, control-constrained, state-constrained and control-and-state constrained problems are analysed. A preconditioned conjugate method for a reduced problem in the control variable is proposed to solve the unconstrained problem, whereas semismooth Newton methods are discussed for the solution of constrained problems. State constraints are treated via a Moreau-Yosida penalization. Convergence is studied for both the continuous problems and the finite dimensional approximations. In the finite dimensional case, we are able to show convergence of the optimization procedures even in the absence of Tikhonov regularization parameter. Computational aspects are also treated and several numerical examples are included to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

We propose a new globalization strategy that can be used in unconstrained optimization algorithms to support rapid convergence from remote starting points. Our approach is based on using multiple points at each iteration to build a sequence of representative models of the objective function. Using the new information gathered from those multiple points, a local step is gradually improved by updating its direction as well as its length. We give a global convergence result and also provide the parallel implementation details accompanied with a numerical study. Our numerical study shows that the proposed algorithm is a promising alternative as a globalization strategy. [ABSTRACT FROM PUBLISHER]

In this paper, in terms of the Fréchet normal cone, we establish exact separation results for finitely many disjoint closed sets in an Asplund space, which supplement the extremal principle and some fuzzy separation theorems. As an application, we provide a new optimality condition for a constraint optimization problem in terms of Fréchet subdifferential and Fréchet normal cone. [ABSTRACT FROM AUTHOR]

We characterize the local upper Lipschitz property of the stationary point mapping and the Karush–Kuhn–Tucker (KKT) mapping for a nonlinear second-order cone programming problem using the graphical derivative criterion. We demonstrate that the second-order sufficient condition and the strict constraint qualification are sufficient for the local upper Lipschitz property of the stationary point mapping and are both sufficient and necessary for the local upper Lipschitz property of the KKT mapping. [ABSTRACT FROM AUTHOR]

We use directional Lipschitz concepts and a minimal time function with respect to a set of directions in order to derive generalized penalization results for Pareto minimality in set-valued constrained optimization. Then, we obtain necessary optimality conditions for maximization in constrained vector optimization in terms of generalized differentiation objects. To the latter aim, we deduce first some enhanced calculus rules for coderivatives of the difference of two mappings. All the main results of this paper are tailored to model directional features of the optimization problem under study. [ABSTRACT FROM AUTHOR]

In this article we devise two iteration schemes for approximating common fixed points of a finite family of nonexpansive mappings and establish the corresponding strong convergence theorem for the sequence generated by any one of our algorithms. Then we apply our results to approximate a solution of the so-called constrained multiple-set convex feasibility fixed point problem for firmly nonexpansive mappings which covers the multiple-set convex feasibility problem in the literature. In particular, our algorithms can be used to approximate the zero point problem of maximal monotone operators, and the equilibrium problem. Furthermore, the unique minimum norm solution can be obtained through our algorithms for each mentioned problem. [ABSTRACT FROM AUTHOR]

The analysis of the principal-agent problem usually requires the classical first-order approach (FOA). However, the validity of the FOA makes restrictive assumptions on the problem under consideration such as the convexity of the distribution function condition. The main aim of this paper is to compute the optimal wages and characterize a closed form solution to the risk-neutral principal-agent problem with limited liability constraints. The development in this paper mainly invokes certain techniques in the semi-infinite programming rather than the FOA. [ABSTRACT FROM AUTHOR]

The notion of quasi-relative interior was introduced by Borwein and Lewis in 1992 and applied for duality results in partially finite convex optimization problems. In the last 10 years, several articles were dedicated to duality results in infinite-dimensional scalar, vector and set-valued optimization problems using this notion. The aim of this paper is to refine and discuss such results. We do this observing that the notion of quasi-relative interior is related to (non-proper) separation of a convex set and some of its elements, then pointing out the relation between the subdifferentiability of a function associated to a set of epigraph type at a certain point and the fact that a corresponding point is not in the quasi-relative interior of the closed convex hull of the set. [ABSTRACT FROM AUTHOR]

In the first part of this article optimization of polyhedral discrete and differential inclusions is considered, the problem is reduced to convex minimization problem and the necessary and sufficient condition for optimality is derived. The optimality conditions for polyhedral differential inclusions based on discrete-approximation problem according to continuous problems are formulated. In particular, boundedness of the set of adjoint discrete solutions and upper semi-continuity of the locally adjoint mapping are proved. In the second part of this article an optimization problem described by convex inequality constraint is studied. By using the equivalence theorem concerning the subdifferential calculus and approximating method necessary and sufficient condition for discrete-approximation problem with inequality constraint is established. [ABSTRACT FROM AUTHOR]

In this article, a modification on Newton's direction for solving problems of unconstrained optimization is presented. As it is known, a major disadvantage of Newton's method is its slowness or non-convergence for the initial point not being close to optima's neighbourhood. The proposed method generally guarantees the decrement of the norm of the gradient or the value of the objective function at every iteration, contributing to the efficiency of Newton's method. The quadratic convergence of the proposed iterative scheme and the enlargement of the radius of convergence area are proved. The proposed algorithm has been implemented and tested on several well-known test functions. [ABSTRACT FROM AUTHOR]

We present an elementary proof of the Lagrange multiplier theorem for optimization problems with equality constraints in normed linear spaces. Most proofs in the literature rely on advanced concepts and results, such as the implicit function theorem and the Lyusternik theorem. By contrast, the proof given in this article employs only basic results from linear algebra, the critical-point condition for unconstrained minima and the fact that a continuous function attains its minimum over a closed ball in the finite-dimensional space. [ABSTRACT FROM AUTHOR]

This article is concerned with the limiting average variance for discrete-time Markov control processes in Borel spaces, subject to pathwise constraints. Under suitable hypotheses we show that within the class of deterministic stationary optimal policies for the pathwise constrained problem, there exists one with a minimal variance. [ABSTRACT FROM AUTHOR]

Li and Sun [D. Li and X.L. Sun, Existence of a saddle point in nonconvex constrained optimization, J. Global Optim. 21 (2001), pp. 39--50; D. Li and X.L. Sun, Convexification and existence of saddle point in a p-th-power reformulation for nonconvex constrained optimization, Nonlinear Anal. 47 (2001), pp. 5611--5622], present the existence of a global saddle point of the p-th power Lagrangian functions for constrained nonconvex optimization, under second-order sufficiency conditions and additional conditions that the feasible set is compact and the global solution of the primal problem is unique. In this article, it is shown that the same results can be obtained under additional assumptions that do not require the compactness of the feasible set and the uniqueness of global solution of the primal problem. [ABSTRACT FROM AUTHOR]