Approximate Optimality Conditions Research Articles

Approximate weak optimality conditions in multiobjective generalized Nash equilibrium problems

A multiobjective generalized Nash equilibrium problem (MGNEP) is a Nash equilibrium problem with constraints that include multiobjective games. We focus in this paper on examining the approximate Karush–Kuhn–Tucker (KKT) conditions for MGNEPs and their impact on the global convergence of algorithms. To begin, we define standard approximate weak KKT (standard-AWKKT) conditions for MGNEPs. We demonstrate that every weak efficient solution meets the standard-AWKKT condition. As these optimality conditions are strong, we propose a new set of approximate weak KKT conditions, specifically tailored for MGNEPs, and show their convergence towards weak KKT points provided a cone-continuity property is satisfied. It is important to note that while these newly introduced approximate weak KKT conditions do not serve as optimality conditions for general MGNEPs, we illustrate that they hold true for the special case of a weak variational equilibrium point in a jointly convex MGNEP. Finally, we propose an enhanced Lagrangian-type algorithm for estimating an approximate weak KKT of an MGNEP and demonstrate its global convergence.

Approximate Optimality Conditions for Nonsmooth Optimization Problems

Approximate Optimality Conditions for Nonsmooth Optimization Problems

Approximate optimality conditions and duality results for non-smooth semi-infinite programming problems

This paper concentrates on investigating a semi-infinite optimization problem with cone constraints. Under the attainment of Abadie constraint qualification, an approximate necessary optimality condition by employing Clarke subdifferential for the semi-infinite optimization problem having cone constraints is developed. A new class of functions namely Q-quasiconvex functions is introduced and in the light of the approximate pseudoconvexity and Q-quasiconvexity assumptions, an approximate sufficient optimality condition is investigated using locally Lipschitz functions. Additionally, we formulate the approximate Wolfe's and approximate Mond-Weir dual problems for the non-smooth semi-infinite optimization problem. Subsequently, duality relations in terms of weak, strong and converse results between the semi-infinite optimization model and the aforementioned dual problems are established under the approximate pseudoconvexity and Q-quasiconvexity assumptions. Moreover, to justify the main results of the paper, numerical examples have been shown at suitable places.

Approximate Subdifferential of the Difference of Two Vector Convex Mappings

This paper deals with the strong approximate subdifferential formula for the difference of two vector convex mappings in terms of the star difference. This formula is obtained via a scalarization process by using the approximate subdifferential of the difference of two real convex functions established by Martinez-Legaz and Seeger, and the concept of regular subdifferentiability. This formula allows us to establish approximate optimality conditions characterizing the approximate strong efficient solution for a general DC problem and for a multiobjective fractional programming problem.

Learning of neural network with optimal control tools

Any learned artificial neural network on a given set of observations represents a function of several variables with vector values or real values. However, in general it is unknown except very simply cases and we have trouble to tell anything about its properties behind very general results received from learning data. In applications, such as medicine, it needs to say not only that on training data we get some error, we have to know that an error is not greater than some ɛ for all data for which we consider the system.It is well known that the learned neural network define a function. However, we check correctness of it only on finite number of observed data. We develop an optimal control approach allowing to find approximation of an unknown function realizing the given observable data, parametrized by a set of controls and defined as ordinary differential equations. Moreover, to measure discrepancy, of the output of the network we define a functional into which we include probability distribution function estimating distribution of the data.We develop a dual dynamic programming ideas to formulate a new optimization problem. We apply it to derive and to prove sufficient approximate optimality conditions for approximate neural network which should work correctly for given ɛ with respect to built functional, on a data different than the set of observations.

Approximate optimality conditions and mixed type duality for semi-infinite multiobjective programming problems involving tangential subdifferentials

The aim of this paper is to study the approximate solution for semi-infinite multiobjective programming problems related to tangential subdifferentials. Under some mild assumptions, we obtain approximate necessary and sufficient optimality conditions of a (weakly) quasi $ \varepsilon $-efficient solution for semi-infinite multiobjective programming problems. We also introduce a mixed type dual model to the semi-infinite multiobjective programming problem, which is different from the one considered in the literature. Finally, we derive some approximate weak, strong and converse duality theorems under the assumptions of generalized convexity. Our results extend and improve the corresponding results in the literature. Some examples are given to illustrate some advantages of our results.

On the Convergence Properties of a Stochastic Trust-Region Method with Inexact Restoration

We study the convergence properties of SIRTR, a stochastic inexact restoration trust-region method suited for the minimization of a finite sum of continuously differentiable functions. This method combines the trust-region methodology with random function and gradient estimates formed by subsampling. Unlike other existing schemes, it forces the decrease of a merit function by combining the function approximation with an infeasibility term, the latter of which measures the distance of the current sample size from its maximum value. In a previous work, the expected iteration complexity to satisfy an approximate first-order optimality condition was given. Here, we elaborate on the convergence analysis of SIRTR and prove its convergence in probability under suitable accuracy requirements on random function and gradient estimates. Furthermore, we report the numerical results obtained on some nonconvex classification test problems, discussing the impact of the probabilistic requirements on the selection of the sample sizes.

On approximate optimality conditions for robust multi-objective convex optimization problems

In this paper, we are interested in the study of approximate optimality conditions for weakly ϵ-efficient solutions to robust multi-objective optimization problems ((RMOP) for short) in view of its associated minimax optimization problem (MMOP). To this end, we first establish the relationship between a weakly ϵ-efficient solution to the problem (RMOP) and an α-solution to the problem (MMOP), where and . Then, we explore the representation of the so-called β-normal set (where is a given parameter) to a closed convex set at some reference point by two methods. At last, by employing the α-subdifferential of the max-function and the obtained representation of the β-normal set, we establish an approximate necessary optimality condition for the problem (RMOP). Moreover, we also give an example to illustrate our results.

Optimality conditions, approximate stationarity, and applications – a story beyond lipschitzness

Approximate necessary optimality conditions in terms of Frechet subgradients and normals for a rather general optimization problem with a potentially non-Lipschitzian objective function are established with the aid of Ekeland’s variational principle, the fuzzy Frechet subdifferential sum rule, and a novel notion of lower semicontinuity relative to a set-valued mapping or set. Feasible points satisfying these optimality conditions are referred to as approximately stationary. As applications, we derive a new general version of the extremal principle. Furthermore, we study approximate stationarity conditions for an optimization problem with a composite objective function and geometric constraints, a qualification condition guaranteeing that approximately stationary points of such a problem are M-stationary, and a multiplier-penalty-method which naturally computes approximately stationary points of the underlying problem. Finally, necessary optimality conditions for an optimal control problem with a non-Lipschitzian sparsity-promoting term in the objective function are established.

Characterizations of Approximate Optimality Conditions for Fractional Semi-Infinite Optimization Problems With Uncertainty

This paper is devoted to the investigation of a class of multi-objective fractional semi-infinite optimization problems with uncertain data. Firstly, a robust counterpart for the uncertain multi-objective optimization problem is introduced in terms of robust optimization method. And following the idea due to Dinkelbach, we associate the robust counterpart with a multi-objective optimization problem. Then, by using a scalarization method, its corresponding scalarization optimization problem is established. The relationship between robust solutions of the multi-objective optimization problem and that of its corresponding scalarization optimization problem is described. Finally, by using a robust type sub-differential constraint qualification, the robust optimality condition for approximate quasi-efficient solutions of the multi-objective fractional optimization problem is established.

Approximate optimality conditions and approximate duality theorems for nonlinear semi-infinite programming problems with uncertainty data

Approximate optimality conditions and approximate duality theorems for nonlinear semi-infinite programming problems with uncertainty data

Sequential approximate weak optimality conditions for multiobjective fractional programming problems via sequential calculus rules for the Brøndsted-Rockafellar approximate subdifferential

Sequential approximate weak optimality conditions for multiobjective fractional programming problems via sequential calculus rules for the Brøndsted-Rockafellar approximate subdifferential

Characterizations of robust ε-quasi optimal solutions for nonsmooth optimization problems with uncertain data

ABSTRACT This paper deals with robust ε-quasi optimal solutions for a class of nonsmooth optimization problems with uncertain data. Under some mild assumptions, we first establish, by using robust optimization (i.e. worst-case) approach, approximate optimality conditions for this uncertain nonsmooth optimization problem. Then, we introduce a Mixed-type robust approximate dual problem of this uncertain optimization problem, and explore their relationships. Moreover, using a scalarization method, we derive optimality conditions for robust weakly approximate efficient solutions for an uncertain nonsmooth multiobjective optimization problem. We also obtain approximate duality theorems for the uncertain nonsmooth multiobjective optimization problem.

Control of blowup and approximate optimality conditions for the wave equations

We study the blowup problems for semilinear wave equations with control function in the domain and on the boundary. Sufficient optimality conditions for controlled blowup time are derived in terms of new dynamic programming methodology. We define ε-optimal value function and we construct sufficient ε-optimal conditions for that function again in terms of new dynamic programming inequality.

Robust approximate optimal solutions for nonlinear semi-infinite programming with uncertainty

In this paper, we deal with robust approximate quasi optimal solutions for a class of nonlinear semi-infinite programming with data uncertainty in both the objective and constraints. By using a new robust-type subdifferential constraint qualification and some generalized convexity assumptions, we first establish approximate optimality conditions for robust approximate quasi optimal solutions of . Then, we introduce a Mixed-type robust approximate dual problem for and investigate robust approximate duality relations between them. Furthermore, we establish a nonsmooth robust approximate saddle point theorem for an uncertain approximate Lagrangian function associated with .

Finite Alternation Theorems and a Constructive Approach to Piecewise Polynomial Approximation in Chebyshev Norm

One of the purposes in this paper is to provide a better understanding of the alternance property which occurs in Chebyshev polynomial approximation and continuous piecewise polynomial approximation problems. In the first part of this paper, we prove that alternating sequences of any continuous function are finite in any given segment and then propose an original approach to obtain new proofs of the well known necessary and sufficient optimality conditions. There are two main advantages of this approach. First of all, the proofs are intuitive and easy to understand. Second, these proofs are constructive and therefore they lead to new alternation-based algorithms. In the second part of this paper, we develop new local optimality conditions for free knot polynomial spline approximation. The proofs for free knot approximation are relying on the techniques developed in the first part of this paper. The piecewise polynomials are required to be continuous on the approximation segment.

Chebyshev Multivariate Polynomial Approximation and Point Reduction Procedure

We apply the methods of nonsmooth and convex analysis to extend the study of Chebyshev (uniform) approximation for univariate polynomial functions to the case of general multivariate functions (not just polynomials). First of all, we give new necessary and sufficient optimality conditions for multivariate approximation, and a geometrical interpretation of them which reduces to the classical alternating sequence condition in the univariate case. Then, we present a procedure for verification of necessary and sufficient optimality conditions that is based on our generalization of the notion of alternating sequence to the case of multivariate polynomials. Finally, we develop an algorithm for fast verification of necessary optimality conditions in the multivariate polynomial case.

Formula omitted] optimality conditions for a finite-time horizon

In this paper we establish the interpolatory model reduction framework for optimal approximation of MIMO dynamical systems with respect to the H2 norm over a finite-time horizon, denoted as the H2(tf) norm. Using the underlying inner product space, we derive the interpolatory first-order necessary optimality conditions for approximation in the H2(tf) norm. Then, we develop an algorithm, which yields a locally optimal reduced model that satisfies the established interpolation-based optimality conditions. We test the algorithm on various numerical examples to illustrate its performance.

Characterizations of Approximate Duality and Saddle Point Theorems for Nonsmooth Robust Vector Optimization

This article deals with robust approximate Pareto efficient solutions for a class of nonsmooth vector optimization problems with data uncertainty. By using robust optimization and scalarization methodology, we first obtain some nonsmooth approximate optimality conditions for this uncertain optimization problem. Then, we introduce a Wolfe-type robust approximate dual problem for this uncertain optimization problem and obtain some nonsmooth robust approximate duality properties. Moreover, we investigate nonsmooth approximate saddle point theorems of an uncertain approximate vector Lagrangian function associated with this uncertain optimization problem.

A Newton-CG algorithm with complexity guarantees for smooth unconstrained optimization

We consider minimization of a smooth nonconvex objective function using an iterative algorithm based on Newton’s method and the linear conjugate gradient algorithm, with explicit detection and use of negative curvature directions for the Hessian of the objective function. The algorithm tracks Newton-conjugate gradient procedures developed in the 1980s closely, but includes enhancements that allow worst-case complexity results to be proved for convergence to points that satisfy approximate first-order and second-order optimality conditions. The complexity results match the best known results in the literature for second-order methods.