Set-valued Constraints Research Articles

Optimal Safety for Constrained Differential Inclusions Using Nonsmooth Control Barrier Functions

For a broad class of nonlinear systems, we formulate the problem of guaranteeing safety with optimality under constraints. Specifically, we define controlled safety for differential inclusions with constraints on the states and the inputs. Through the use of nonsmooth analysis tools, we show that a continuous optimal control law can be selected from a set-valued constraint capturing the system constraints and conditions guaranteeing safety using control barrier functions. Our results guarantee optimality and safety via a continuous state-feedback law designed using nonsmooth control barrier functions. An example pertaining to obstacle avoidance with a target illustrates our results and the associated benefits of using nonsmooth control barrier functions.

External polyhedral estimates of reachable sets of discrete-time systems with integral bounds on additive terms

<p style='text-indent:20px;'>We deal with the reachability problem for linear and bilinear discrete-time uncertain systems under integral non-quadratic constraints on additive input terms and set-valued constraints on initial states. The bilinearity is caused by an interval type uncertainty in coefficients of the system. Algorithms for constructing external parallelepiped-valued (shorter, polyhedral) estimates of reachable sets are presented. For linear time-invariant systems, two techniques for constructing touching external estimates with constant orientation matrices are described and compared. For time-dependant bilinear systems, parallelepiped-valued estimates are constructed. For bilinear systems with constant coefficients, nonconvex estimates are proposed in the form of unions of parallelepipeds. Evolution of all estimates is determined by systems of recurrence relations.</p>

First- and second-order optimality conditions in optimistic bilevel set-valued programming

ABSTRACT In this work we adapt the main results from Khanh and Tung [First and second-order optimality conditions without differentiability in multivalued vector optimization. Positivity. 2015;19:817–841] for a general set-valued optimization problem to an optimistic bilevel set-valued programming problem as an optimization problem with implicitly given set-valued constraints. Using the optimal value function, we convert our problem into a one level set-valued optimization problem with general inequality constraints and derive both necessary conditions and sufficient conditions of order one and two. The main tools we exploit are approximations of set-valued mappings. To illustrate the obtained results, some examples are given.

Comment

Comment

Optimality and duality in set-valued optimization utilizing limit sets

Abstract This paper deals with optimality conditions and duality theory for vector optimization involving non-convex set-valued maps. Firstly, under the assumption of nearly cone-subconvexlike property for set-valued maps, the necessary and sufficient optimality conditions in terms of limit sets are derived for local weak minimizers of a set-valued constraint optimization problem. Then, applications to Mond-Weir type and Wolfe type dual problems are presented.

Necessary optimality conditions for optimistic bilevel programming problems using set-valued programming

In this paper we adapt the main results from Amahroq and Gadhi (J Glob Optim 21:435---443, 2001) for a general set-valued optimization problem to an optimistic bilevel programming problem as an optimization problem with implicitly given set-valued constraint. Since this constraint is assumed to be upper but not lower semicontinuous in the sense of Berge, we need to deal with a lower semicontinuous distance function to this mapping. In order to approximate the gradient of the distance function, we introduce a new concept for a directional convexificator. Some calculus rules for this new tool are adapted and several properties are characterized. The main result presents optimality conditions for an optimistic bilevel programming problem using a convexificator constructed with the aid of the directional convexificator.

The existence of solutions for dynamic inclusions on time scales via duality

The problem of existence of solutions to nabla differential equations and nabla differential inclusions on time scales is considered. Under a special form of the set-valued constraint map, sufficient conditions for the existence of at least one solution, that stay in the constraint set, are derived.

Solving set-valued constraint satisfaction problems

In this paper, we consider the resolution of constraint satisfaction problems in the case where the variables of the problem are subsets of $${\mathbb{R}^{n}}$$ . In order to use a constraint propagation approach, we introduce set intervals (named i-sets), which are sets of subsets of $${\mathbb{R}^{n}}$$ with a lower bound and an upper bound with respect to the inclusion. Then, we propose basic operations for i-sets. This makes possible to build contractors that are then used by the propagation to solve problem involving sets as unknown variables. In order to illustrate the principle and the efficiency of the approach, a testcase is provided.

New Generalized Convexity Notion for Set-Valued Maps and Application to Vector Optimization

In this paper, we introduce a new generalized convexity notion for set-valued maps, called ic-cone-convexlikeness, and use it as the main tool to derive an alternative theorem and necessary conditions for efficient, weakly efficient, and Benson properly efficient solutions of the problem of minimizing a set-valued map subject to set-valued constraints. Our results are valid for a class of optimization problems broader than that of the problems considered in Refs. 1--6 and generalize the corresponding results of these references.

A Measure-Valued Approach to Convex Set-Valued Dynamics

In this paper we introduce a new concept of differentiable set-valued dynamics and show the existence of solution curves. To do this we introduce simple set-valued functions which are completely determined by a convex set and a function on the boundary of this set, and show the differentiability of these functions from the right at t=0. Then we approximate local solutions of our dynamical system by set-valued functions consisting of a finite number of simple functions. The dynamics are described by functions on the boundaries of convex bodies. (This is analogous to the description of single-valued dynamics by vector fields.) It can be shown that in some special cases applicable to set-valued constraint stochastic optimization, the local solutions can be glued together to provide global ones. A constraint stochastic optimization problem is investigated.

Optimum Problems with Measurable Set-Valued Constraints

In this paper, we provide a complete analysis of second-order admissible variations to inequality-type constraints, which are given in terms of measurable set-valued functions whose images are closed convex sets with nonempty interior. As an application, we consider optimization problems where such constraints are present, and we deduce second-order necessary conditions for optimality.

Special issue on `Set-valued optimization': Editors' preface

This special issue on Set-Valued Optimization treats set-valued analysis and its application to optimization. On the one hand optimization problems are investigated with constraints and/or an objective function described by setvalued maps, and on the other hand investigations in set-valued analysis are applied to standard optimization problems. In the last decade there has been an increasing interest in set-valued optimization. General optimization problems with set-valued constraints or a set-valued objective function are closely related to problems in stochastic programming, fuzzy programming and optimal control. If the values of a given function vary in a speci®ed region, this fact could be described using a membership function in the theory of fuzzy sets or using information on the distribution of the function values. In this general setting probability distributions or membership functions are not needed because only sets are considered. Optimal control problems with di erential inclusions belong to this class of set-valued optimization problems as well. Set-valued optimization seems to have the potential to become a bridge between di erent areas in optimization. And it is a substantial extension of standard optimization theory. Set-valued analysis is the most important tool for such an advancement in continuous optimization. And conversely, the development of set-valued analysis receives important impulses from optimization. This issue makes evident the interplay between set-valued analysis and optimization. The articles in this special issue throw light on various facets of set-valued analysis in optimization. These contributions re ect scienti®c activities in Europe, China, Australia and North America. The papers may be divided into two groups:

Optimum Problems with Certain Lower Semicontinuous Set-Valued Constraints

In this paper, we provide a complete analysis of second-order admissible variations to inequality type constraints that are given in terms of lower semicontinuous set-valued functions whose images are closed convex sets with nonempty interior. As an application, we consider optimization problems where such constraints are present and we deduce second-order necessary conditions for optimality.

On Generalized Gradients in Optimization Problems with Set-Valued Constraints

We investigate the value function f(x) = inf φ (x, y) where the infimum is over all y ∈ a(x) for some given set-valued map a. Under specified conditions we provide an inner approximation for its generalized gradient. In some cases a full description of this generalized gradient is given. The results are of use in numerical solution of various optimization problems. Illustrative examples are given.