Transformation of Quadratic Forms to Perfect Squares for Broken Extremals

In this paper we study a quadratic form which corresponds to an extremal with piecewise continuous control in variational problems. This form, compared with the classical one, has some new terms connected with the set of all points of discontinuity of the control. Its positive definiteness is a sufficient optimality condition for broken extremals. We show that if there exists a solution to corresponding Riccati equation satisfying some jump condition at each point of the set , then the quadratic form can be transformed to a perfect square, just as in the classical case. As a result we obtain sufficient conditions for positive definiteness of the quadratic form in terms of the Riccati equation and hence, sufficient optimality conditions for broken extremals.     

The Integrability Conditions in the Inverse Problem of the Calculus of Variations for Second-Order Ordinary Differential Equations

A novel approach to a coordinate-free analysis of the multiplier question in the inverseproblem of the calculus of variations, initiated in a previous publication, is completed in thefollowing sense: under quite general circumstances, the complete set of passivity or integrabilityconditions is computed for systems with arbitrary dimension n. The results are appliedto prove that the problem is always solvable in the case that the Jacobi endomorphism of thesystem is a multiple of the identity. This generalizes to arbitrary n a result derived byDouglas for n=2.     

关于变分学中逆问题的研究

本文研究了变分学中的逆问题.通过变积概念的引入,给出了系统地研究变分学中逆问题的一种新途径.将这种方法应用于线弹性动力学和粘性流体力学中,建立了各自的变分原理和广义变分原理.     

Duality Bound Method for the General Quadratic Programming Problem with Quadratic Constraints

The purpose of this article is to develop a branch-and-bound algorithm using duality bounds for the general quadratically-constrained quadratic programming problem and having the following properties: (i) duality bounds are computed by solving ordinary linear programs; (ii) they are at least as good as the lower bounds obtained by solving relaxed problems, in which each nonconvex function is replaced by its convex envelope; (iii) standard convergence properties of branch-and-bound algorithms for nonconvex global optimization problems are guaranteed. Numerical results of preliminary computational experiments for the case of one quadratic constraint are reported.     

An Alternative Method for a Classical Problem in the Calculus of Variations

A numerical technique for determining the solution of the brachistochrone problem is presented. The brachistochrone problem is first formulated as a non-linear optimal control problem. Using Chebyshev nodes, we construct the Mth degree polynomial interpolation to approximate the state and the control variables. Application of this method results in the transformation of differential and integral expressions into some non-linear algebraic equations to which Newton-type methods can be applied. Simulation studies demonstrate computational advantages relative to existing methods in the literature.     

Addendum to: The Integrability Conditions in the Inverse Problem of the Calculus of Variations for Second-Order Ordinary Differential Equations

An omission in the outline of the general approach to the inverse problem in Acta Appl. Math. (54 (1998), pp. 233–273) is clarified.     

Quadratic Functionals with General Boundary Conditions

The purpose of this paper is to give the Reid ``Roundabout Theorem' for quadratic functionals with general boundary conditions. In particular, we describe the so-called coupled point and regularity condition introduced in [16] in terms of Riccati equation solutions. Accepted 27 February 1996     

The Inverse Problem of the Calculus of Variations: Separable Systems

This paper deals with the inverse problem of the calculus of variations for systems of second-order ordinary differential equations. The case of the problem which Douglas, in his classification of pairs of such equations, called the separated case is generalized to arbitrary dimension. After identifying the conditions which should specify such a case for n equations in a coordinate-free way, two proofs of its variationality are presented. The first one follows the line of approach introduced by some of the authors in previous work, and is close in spirit, though being coordinate independent, to the Riquier analysis applied by Douglas for n = 2. The second proof is more direct and leads to the discovery that belonging to the separated case has an intrinsic meaning for the given second-order differential equations: the system is separable in the sense that it can be decoupled into n pairs of first-order equations.     

Global Optimization Techniques for Solving the General Quadratic Integer Programming Problem

We consider the problem of minimizing a general quadratic function over a polytope in the n-dimensional space with integrality restrictions on all of the variables. (This class of problems contains, e.g., the quadratic 0-1 program as a special case.) A finite branch and bound algorithm is established, in which the branching procedure is the so-called integral rectangular partition, and the bound estimation is performed by solving a concave programming problem with a special structure. Three methods for solving this special concave program are proposed.     

A Planning Problem Combining Calculus of Variations and Optimal Transport

We consider some variants of the classical optimal transport where not only one optimizes over couplings between some variables x and y but also over some control variables governing the evolutions of these variables with time. Such a situation is motivated by an assignment problem of tasks with workers whose characteristics can evolve with time (and be controlled). We distinguish between the coupled and decoupled case. The coupled case is a standard optimal transport with the value of some optimal control problem as cost. The decoupled case is more involved since it is nonlinear in the transport plan.     

Normality of Necessary Optimality Conditions for Calculus of Variations Problems with State Constraints

Set-Valued and Variational Analysis - We consider non-autonomous calculus of variations problems with a state constraint represented by a given closed set. We prove that if the interior of the...     

Inverse Problem of the Calculus of Variations for Systems of Differential-Difference Equations of Second Order

The inverse problem of the calculus of variations for second-order nonlinear and linear systems of differential-difference equations is considered. The relationship between the formal potentiality of a linear system with constant coefficients and the parity of its characteristic function is established.     

Extremality in Solving General Quasilinear Parabolic Inclusions

The paper deals with an initial boundary-value problem for a parabolic inclusion whose multivalued term has the structure of a difference between the Clarke generalized gradient of some locally Lipschitz function verifying a unilateral growth condition and the subdifferential of a convex function, and where the elliptic part is expressed by a general quasilinear operator of the Leray-Lions type. Our results address not only the existence of solutions, but also the extremality inside an order interval determined by appropriately defined upper and lower solutions as well as the compactness of the solution set in suitable spaces.     

Adjoint Coexhausters in Nonsmooth Analysis and Extremality Conditions

In the classical (“smooth”) mathematical analysis, a differentiable function is studied by means of the derivative (gradient in the multidimensional space). In the case of nondifferentiable functions, the tools of nonsmooth analysis are to be employed. In convex analysis and minimax theory, the corresponding classes of functions are investigated by means of the subdifferential (it is a convex set in the dual space), quasidifferentiable functions are treated via the notion of quasidifferential (which is a pair of sets). To study an arbitrary directionally differentiable function, the notions of upper and lower exhausters (each of them being a family of convex sets) are used. It turns out that conditions for a minimum are described by an upper exhauster, while conditions for a maximum are stated in terms of a lower exhauster. This is why an upper exhauster is called a proper one for the minimization problem (and an adjoint exhauster for the maximization problem) while a lower exhauster will be referred to as a proper one for the maximization problem (and an adjoint exhauster for the minimization problem). The directional derivatives (and hence, exhausters) provide first-order approximations of the increment of the function under study. These approximations are positively homogeneous as functions of direction. They allow one to formulate optimality conditions, to find steepest ascent and descent directions, to construct numerical methods. However, if, for example, the maximizer of the function is to be found, but one has an upper exhauster (which is not proper for the maximization problem), it is required to use a lower exhauster. Instead, one can try to express conditions for a maximum in terms of upper exhauster (which is an adjoint one for the maximization problem). The first to get such conditions was Roshchina. New optimality conditions in terms of adjoint exhausters were recently obtained by Abbasov. The exhauster mappings are, in general, discontinuous in the Hausdorff metric, therefore, computational problems arise. To overcome these difficulties, the notions of upper and lower coexhausters are used. They provide first-order approximations of the increment of the function which are not positively homogeneous any more. These approximations also allow one to formulate optimality conditions, to find ascent and descent directions (but not the steepest ones), to construct numerical methods possessing good convergence properties. Conditions for a minimum are described in terms of an upper coexhauster (which is, therefore, called a proper coexhauster for the minimization problem) while conditions for a maximum are described in terms of a lower coexhauster (which is called a proper one for the maximization problem). In the present paper, we derive optimality conditions in terms of adjoint coexhausters.     

The Inverse Problem of the Calculus of Variations for Sixth- and Eighth-order Scalar Ordinary Differential Equations

On the equation manifold of the 2nth-order scalar ordinary differential equation, n3,