Constrained Extremum Problems Research Articles

Extremum Problem Formulations of Mixed-Form Models for Elastostatics

Expression is given here to the general elastostatics problem in the form of convex constrained extremum problems. The alternate formulations make use of a decomposition of the measure of stress and strain into two variationally independent components. This provides for the interpretation of the results as mixed models, as is to be demonstrated in what follows

Coercivity and image of constrained extremum problems

The concept of coercivity is extended to the image space, and it is exploited to obtain a lower bound for the minimum of a constrained extremum problem and a sufficient condition for the optimality. Problems which are coercive in the image space turn out to have zero duality gap. A possible application to variational inequalities is illustrated by means of an example.

Regularity conditions for constrained extremum problems via image space

Exploiting the image-space approach, we give an overview of regularity conditions. A notion of regularity for the image of a constrained extremum problem is given. The relationship between image regularity and other concepts is also discussed. It turns out that image regularity is among the weakest conditions for the existence of normal Lagrange multipliers.

A general scheme for first order approximations in optimization

In this paper we present a general approach for treating first order approximation schemes constrained extremum problems. We underline that, roughly speaking, these schemes present two geometric one and an analytical one. We will show that separation theorems and tangent fundamental role for deriving a scheme from the geometrical viewpoint; from the other derivative is the key concept for the analytical viewpoint. The purpose of the paper is scheme for first order approximations in optimization and to put in evidence the connections geometrical and analytical properties of the problem. The main interest is devoted to those which are useful for deriving necessary optimality conditions and for the construction of Solution methods.

An extension of the Hestenes necessary condition

After the pioneering approach of Hestenes, which has not been sufficiently considered, the concept of image of a constrained extremum problem has been recognized recently to be a powerful tool in optimization. Hence in this paper, we consider a necessary optimality condition for a nondifferentiable extremum problem established by Hestenes in the image space and, by again exploiting the concept of image of the problem, a necessary condition embodying that of Hestenes is given. Moreover, the study is completed by expressing the previous result in the space where the unknowns of the given problem run. In this way, we obtain a Lagrangian-type necessary condition holding for a particular class of semidifferentiable problems.

The second derivative test for constrained extremum problems

The method of Lagrange multipliers is derived from a geometric point of view, with the emphasis on a second derivative test within this context. Although the derivation is fairly detailed, the statement of the results is succinct and easily explained in the classroom. Examples are given in the Appendix.

On a Lagrangian sufficient optimality condition

By exploiting the concept of image of a constrained extremum problem, a sufficient optimality condition of Lagrangian type is established. The result is achieved by applying the idea of coercivity in the image space; this produces new results with respect to the use of coercivity in the given space. The condition embeds a known one.

A necessary optimality condition for nondifferentiable constrained extremum problems

The purpose of this paper is to establish a necessary optimality condition for a certain class of nondifferentiable extremum problems. The analysis is developed via image-space approach and the result generalizes, in finite-dimensional spaces, the classical results concerning locally Lipschitz mathematical programming.

Semidifferentiable functions and necessary optimality conditions

In the last two decades, there has been an increasing interest in nonsmooth optimization, both from a theoretical viewpoint and because of several applications. Necessary optimality conditions, as well as other important topics, have received new attention (see, for instance, Refs. 1–11 and references therein). In recent papers (see, for instance, Refs. 12–18 and references therein), theorems of the alternative for generalized systems have been studied and their use in optimization has been exploited. As a consequence of this analysis, the concept of image of a constrained extremum problem has been developed; such a concept, whose introduction goes back to the work of Caratheodory, has only recently been recognized to be a powerful tool (Refs. 8, 10, 13, 14, 18–21). On the basis of these ideas, in the present paper we deal with a necessary condition for constrained extremum problems having a finite-dimensional image, while those having an infinite-dimensional one will be treated in a subsequent paper. The necessary condition is established within a class of semidifferentiable functions, which is introduced here and which embraces several classic types of functions (e.g., convex functions, differentiable functions, and even some discontinuous functions). The condition embodies the classic theorems of Lagrange, John, Karush, Kuhn-Tucker, and Euler.

On the image of a constrained extremum problem and some applications to the existence of a minimum

A new approach to constrained optimization, which has appeared recently under various forms and in several contexts, is presented in a general and unifying setting. This approach is then employed to establish some new conditions for the existence of the minimum of a constrained minimum problem.

Theorems of the alternative for multifunctions with applications to optimization: General results

Theorems of the alternative and separation theorems have been shown to be very useful concepts in constrained extremum problems (see, for instance, Refs. 1–12). Their use has stressed the concept of image of a constrained extremum problem, which has turned out to be a powerful and promising tool for investigating the main aspects of optimization (see Refs. 13 and 19). It should be pointed out that, in this approach, a finite-dimensional image problem can be associated to the given extremum problem, even if this is infinite-dimensional and provided that its constraints are expressed by functionals. Such a development can be carried on by means of theorems of the alternative for systems of single-valued functions. In this paper, theorems of the alternative for systems of multifunctions are studied, some general properties are stated, and connections with known results investigated. It is shown how the present approach can be used to analyze extremum problems, where the image of the domain of the constraining functions belongs to a functional space. Such a development will be carried on in a subsequent paper.

Regularity conditions for constrained extremum problems

Necessary and/or sufficient conditions are stated in order to have regularity for nondifferentiable problems or differentiable problems. These conditions are compared with some known constraint qualifications.

On minimization of a function under an equality chance constraint

The joint chance constrained extremum problem is considered. Basing on independent realizations of a random vector the Parzen kernel-type estimate of the constraint function is proposed. The initial problem and the problem where the constraint function is replaced by this estimate are compared.

Structure of the Correspondence Principle at an Extremum Point

One of the most powerful principles of microeconomics is that comparative static properties of a given behavioural equation can often be derived from the postulate that it is a solution function of some extremum problem. Applying this principle, economists have individually analyzed a wide variety of specific economic models-usually through complicated derivations involving bordered Hessians. The main purpose of the present article is to derive general formulae of comparative statics for the extremum problem with many constraint functions, where the same shift parameters appear in both target and constraint functions and all of the constraint and target functions are non-linear. This problem contains most of the models whose comparative statics have been studied so far. Our method of comparative statics is related to those proposed by Houthakker (1951-52), Samuelson (1960, 1965), Diamond and McFadden (1974), Silberberg (1974) and Gorman (1976). Threefeatures distinguish the present approach from those ofthis literature. First, our method yields satisfactory comparative static conditions when constraints contain parameters. In particular, they contain the theorems of recent applied economic models whose shift parameters appear in both target and constraint functions.' Second, only unconstrained extremum conditions are used in our derivation of comparative statics for constrained extremum problems; the Lagrangean multiplier method is dispensed with as an analytical tool. Third, the present framework yields heuristic proofs of both the Lagrangean method and the Kuhn-Tucker conditions. The method may be outlined as follows. We transform the original constrained extremum problem into an unconstrained extremum problem of the associated gain function. This function contains as the variables vectors of both decision variables and parameters of the original problem. The unconstrained extremum conditions of the gain function with respect to the decision variables establish the Lagrangean method, while those with respect to the parameter vector yield the comnparative static formulae. Our method of derivation may be called the gain function method. In the present paper we will nowhere assume convexity, nor will we employ theories on convex sets and functions. The next section illustrates the gain function approach in the simplest possible situation by proving the Lagrangean method. Comparative static results are derived in Section 2. The Le Chatelier-Samuelson Principle is generalized in Section 3. The gain function method is applied to maximization problems with inequality constraints in Section 4. The final section presents economic applications of the results.

Existence of solutions of pseudoconcave inequalities

Tucker's results for systems of linear inequalities are extended to nonlinear systems. Pseudoconcave or homogeneous functions are covered. The proof is performed through a simple, constrained extremum problem. Some applications are also suggested.

A method of weakening the constraints in constrained extremum problems

A method of weakening the constraints in constrained extremum problems

Dual and inverse formulations of constrained extremum problems

In this paper we consider constrained extremum problems of th form Pp:infu∈t−1(p)f(u) where f and t are continuously differentiable functionals on a reflexive Banach space V and where t-1(p) denotes the level set of the functional t with value p ϵ R. Related to problems Pp we investigate inverse extremum problems, which are extremum problems for the functional t on level sets of the functional f. Under conditions that guarantee the existence of solutions of Pp, let h(p) denote the value of f at such a solution. If h is a (locally) convex function at some p̄ ϵ R, we show that it is possible to define a dual problem of Pp̄. This dual problem is a saddle-point formulation for the functional V × RЭ(u,μ)→f(u)−μ[t(u)−p̄]: for some extreme value μ̄ (which is the Lagrange multiplier of a solution of Pp̄) the solutions of Pp̄ are precisely the (local) minimal points of the functional f − μ̄t on V. It is shown how these results can be used to describe solution branches of nonlinear eigenvalue problems (of semilinear elliptic type) with a global parameter, such as p ϵ R, instead of with the eigenvalue as a parameter.

Continuation of solutions of constrained extremum problems and nonlinear eigenvalue problems

In this paper we continue our investigations, begun in the previous paper, of describing the solution sets of a constrained extremum problems inf f(u), uϵt−1(p) where f and t are twice continuously differentiable functionals on a reflexive Banach space V, and t-1(p) denotes the level set of the functional t with value p ϵ R. Considering p as a parameter in (∗) we obtain results concerning the continuation of solutions of (∗) and consequently also concerning specific solution branches of the nonlinear eigenvalue problem f′(u) = μt′(u) The general results are applied to functionals which lead to nonlinear eigenvalue problems of a semilinear elliiptic type and in particular we cosider a specific example for which there occurs “bending” of a solution curve (u,μ) of (∗∗).

Methods for solving constrained extremum problems in the presence of random noise

THE EXTREMUM problem with equation-type constraints is considered, when the measurements of all functions and their gradients are subject to noise. Three methods of solution are described: they are modifications of the method of Lagrange multipliers, the method of penalty functions, and the method of penalty estimates respectively. The methods are shown to be convergent in a specific probability sense.

Demand Theory: Time Allocation and Outdoor Recreation

The concept of demand has evolved through the centuries, enriched by the natural-value, exchange-value controversy and the diamond, water paradox of the classicists [11], the intuitive insights of Marshall [8], and the mathematical rigor of Slutsky [10], and Hicks and Men [6]. These developments led to the conceptualization of a demand function as a solution function to a constrained extremum problem [5,6,9, 10].