Search

The scalar finite-dimensional case has been discussed in the first part of this work series, which aims at exploiting the image space analysis to establish a general regularity condition for constrained extremum problems. Based on this preliminary result, the present paper dedicates itself to further study the regularity conditions for vector constrained extremum problems in a Euclidean space. The case of infinite-dimensional image will be the subject of a subsequent paper. [ABSTRACT FROM AUTHOR]

Image space analysis is a new tool for studying scalar and vector constrained extremum problems as well as generalized systems. In the last decades, the introduction of image space analysis has shown that the image space associated with the given problem provides a natural environment for the Lagrange theory of multipliers and that separation arguments turn out to be a fundamental mathematical tool for explaining, developing and improving such a theory. This work, with its 3 parts, aims at contributing to describe the state-of-the-art of image space analysis for constrained optimization and to stress that it allows us to unify and generalize the several topics of optimization. In this 1st part, after a short introduction of such an analysis, necessary and sufficient optimality conditions are treated. Duality and penalization are the contents of the 2nd part. The 3rd part deals with generalized systems, in particular, variational inequalities and Ky Fan inequalities. Some further developments are discussed in all the parts. [ABSTRACT FROM AUTHOR]

In this paper, by virtue of the image space analysis, the general scalar robust optimization problems under the strictly robust counterpart are considered, among which, the uncertainties are included in the objective as well as the constraints. Besides, on the strength of a corrected image in a new type, an equivalent relation between the uncertain optimization problem and its image problem is also established, which provides an idea to tackle with minimax problems. Furthermore, theorems of the robust weak alternative as well as sufficient characterizations of robust optimality conditions are achieved on the frame of the linear and nonlinear (regular) weak separation functions. Moreover, several necessary and sufficient optimality conditions, especially saddle point sufficient optimality conditions for scalar robust optimization problems, are obtained. Finally, a simple example for finding a shortest path is included to show the effectiveness of the results derived in this paper. [ABSTRACT FROM AUTHOR]

This note aims at stimulating research on set-valued optimization through image space analysis. [ABSTRACT FROM AUTHOR]

In Part I, sufficient and necessary optimality conditions and the image regularity conditions of constrained scalar and vector extremum problems are reviewed for Image Space Analysis. Part II presents the main feature of the duality and penalization of constrained scalar and vector extremum problems by virtue of Image Space Analysis. In the light, as said in Part I and Part II, to describe the state of Image Space Analysis for constrained optimization, and to stress that it allows us to unify and generalize the several topics of Optimization, in this Part III, we continue to give an exhaustive literature review on separation functions, gap functions and error bounds for generalized systems. Part III also throws light on some research gaps and concludes with the scope of future research in this area. [ABSTRACT FROM AUTHOR]

By means of the image space analysis, a necessary and sufficient condition is established for the existence of vector extrema. In the scalar case, the sufficient part is shown to shrink to a known one. [ABSTRACT FROM AUTHOR]

Image space analysis has proved to be instrumental in unifying several theories, apparently disjoint from each other. With reference to constraint qualifications/regularity conditions in optimization, such an analysis has been recently introduced by Moldovan and Pellegrini. Based on this result, the present paper is a preliminary part of a work, which aims at exploiting the image space analysis to establish a general regularity condition for constrained extremum problems. The present part deals with scalar constrained extremum problems in a Euclidean space. The vector case as well as the case of infinite-dimensional image will be the subject of a subsequent part. [ABSTRACT FROM AUTHOR]

Based on recent results on image space analysis, the paper aims at describing a fixed point approach to vector optimization problems. Possible extensions to the bi-level vector optimization are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we establish a group of closed-form formulas for the maximal and minimal ranks of a nonlinear matrix expression with respect to two variant matrices by using a linearization method and some known formulas for extremal ranks of linear matrix expressions. In addition, by using some pure algebraic operations of matrices and their generalized inverses, we derive the maximal and minimal ranks of the above nonlinear matrix expression, where the two variant matrices are any solutions of two consistent matrix equations. As an application, we derive some sufficient and necessary conditions for the existence of the solution of a nonlinear matrix function. [ABSTRACT FROM AUTHOR]

This article is devoted to the numerical computation of distributed null controls for the 1D heat equation. The goal is to compute a control that drives (a numerical approximation of) the solution from a prescribed initial state exactly to zero. We extend the earlier contribution of Carthel, Glowinski, and Lions, which is devoted to the computation of controls of minimal square-integrable norm. We start from some constrained extremal problems (involving unbounded weights in time), introduced by Fursikov and Imanuvilov, and we apply appropriate duality techniques. Then, we provide numerical approximations of the associated dual problems, and apply conjugate gradient algorithms. Finally, several experiments are presented, and we highlight the influence of the weights and analyze this approach in terms of robustness and efficiency. Also, the results are compared with others in a previous article of the authors, where primal methods were considered. [ABSTRACT FROM AUTHOR]

The Graph-Lagrangian of a hypergraph has been a useful tool in hypergraph extremal problems. In most applications, we need an upper bound for the Graph-Lagrangian of a hypergraph. Frankl and Füredi conjectured that the $${r}$$ -graph with $$m$$ edges formed by taking the first $$\textit{m}$$ sets in the colex ordering of the collection of all subsets of $${\mathbb N}$$ of size $${r}$$ has the largest Graph-Lagrangian of all $$r$$ -graphs with $$m$$ edges. In this paper, we show that the largest Graph-Lagrangian of a class of left-compressed $$3$$ -graphs with $$m$$ edges is at most the Graph-Lagrangian of the $$\mathrm 3 $$ -graph with $$m$$ edges formed by taking the first $$m$$ sets in the colex ordering of the collection of all subsets of $${\mathbb N}$$ of size $${3}$$ . [ABSTRACT FROM AUTHOR]

We consider a time-optimal problem for the Reeds and Shepp model describing a moving point on a plane, with a onesided variation of the speed and a free final direction of the velocity. Using the Pontryagin Maximum Principle, we obtain all possible types of extremal and, analyzing them and discarding nonoptimal ones, construct the optimal synthesis. [ABSTRACT FROM AUTHOR]

This paper studies optimal control problems and sub-Riemannian geometry on a nonholonomic macroeconomic system. The main results show that a nonholonomic macroeconomic system is controllable either by trajectories of a single-time driftless control system (single-time bang-bang controls), or by nonholonomic geodesics or by sheets of a two-time driftless control system (two-time bang-bang controls). They are strongly connected to the possibility of describing a nonholonomic macroeconomic system via a Gibbs-Pfaff equation or by four associated vector fields, based on a contact structure of the state space and our isomorphism between thermodynamics and macroeconomics that praises three laws of a nonholonomic macroeconomic system. [ABSTRACT FROM AUTHOR]

This article investigates extremality, stationarity, and regularity properties of infinite collections of sets in Banach spaces. Our approach strongly relies on the machinery developed for finite collections. When dealing with an infinite collection of sets, we examine the behavior of its finite subcollections. This allows us to establish certain primal-dual relationships between the stationarity/regularity properties some of which can be interpreted as extensions of the Extremal principle. Stationarity criteria developed in the article are applied to proving intersection rules for Fréchet normals to infinite intersections of sets in Asplund spaces. [ABSTRACT FROM AUTHOR]

We present a modification of the DIRECT (DIviding RECTangles) algorithm, called DIRECT-G, to solve a box-constrained global optimization problem arising in the detection of gravitational waves emitted by coalescing binary systems of compact objects. This is a hard problem, since the objective function is highly nonlinear and expensive to evaluate, has a huge number of local extrema and unavailable derivatives. DIRECT performs a sampling of the feasible domain over a set of points that becomes dense in the limit, thus ensuring the everywhere dense convergence; however, it becomes ineffective on significant instances of the problem under consideration, because it tends to produce a uniform coverage of the feasible domain, by oversampling regions that are far from the optimal solution. DIRECT has been modified by embodying information provided by a suitable discretization of the feasible domain, based on the signal theory, which takes into account the variability of the objective function. Numerical experiments show that DIRECT-G largely outperforms DIRECT and the grid search, the latter being the reference algorithm in the astrophysics community. Furthermore, DIRECT-G is comparable with a genetic algorithm specifically developed for the problem. However, DIRECT-G inherits the convergence properties of DIRECT, whereas the genetic algorithm has no guarantee of convergence. [ABSTRACT FROM AUTHOR]

Analytic solutions for optimal collision avoidance strategies are of great importance when setting and validating air traffic rules and as a benchmark when validating automated proximity management and collision avoidance systems. Such a solution for optimal air collision avoidance strategies for a coplanar cooperative encounter between two identical aircraft (or ships) was first presented by Merz (Proc. Joint Automatic Control Conf., Pap. 15-3:449–454, ; Navigation 20(2):144–152, ). Unfortunately, Merz provided only a very brief indicative justification for his solution. This paper presents a rigorous analysis of the problem. New results include a characterization of a complete set of extremals, justification for optimal strategies and an analysis of the properties of the regions of different optimal strategies. A simple, practical and sufficiently accurate closed form approximation for dispersal curves that partition the plane of initial positions into the regions of different optimal strategies is also presented. [ABSTRACT FROM AUTHOR]

In this note, we contrast two transformation-based methods to deduce absolute extrema and the corresponding extremizers. Unlike variation-based methods, the transformation-based methods of Carlson and Leitmann and the recent one of Silva and Torres are direct in that they permit obtaining solutions by inspection. [ABSTRACT FROM AUTHOR]

In this note, we derive first and second-order necessary (resp. sufficient) conditions for local minimum (resp. strict local minimum) points of optimization problems governed by set-valued maps. This allows us to present and to solve a generalized Lagrange claim in a multivalued setting. [ABSTRACT FROM AUTHOR]

We use the concept of generalized inverses to show that the extrema of the function G(x) = ∥Ax - b∥2 are minima, A being a rectangular matrix not necessarily of full rank. [ABSTRACT FROM AUTHOR]

First-order and second-order necessary conditions of optimality for an impulsive control problem that remain informative for abnormal control processes are presented and derived. One of the main features of these conditions is that no a priori normality assumptions are required. This feature follows from the fact that these conditions rely on an extremal principle which is proved for an abstract minimization problem with equality constraints, inequality constraints, and constraints given by an inclusion in a convex cone. Two simple examples illustrate the power of the main result. [ABSTRACT FROM AUTHOR]

It is shown that, if a parametrized fämily of extremals can be stratified in a way compatible with the flow map generated by F, then those trajectories of the family which realize the minimal values of the cost in F are indeed optimal in comparison with all trajectories which lie in the region R covered by the trajectories of F. It is not assumed that F is a field covering the state space injectively. As illustration, an optimal synthesis is constructed for a system where the flow of extremals exhibits a simple cusp singularity. [ABSTRACT FROM AUTHOR]