A probabilistic approach to second order variational inequalities with bilateral constraints

We study a class of second order variational inequalities with bilateral constraints. Under certain conditions we show the existence of aunique viscosity solution of these variational inequalities and give a stochastic representation to this solution. As an application, we study a stochastic game with stopping times and show the existence of a saddle point equilibrium.     

On variational problems involving higher order derivatives

Fritz John, and Karush-Kuhn-Tucker type optimality conditions for a constrained variational problem invoving higher order derivatives are obtained. As an application of these Karush-Kuhn-Tucker type optimality conditions, Wolfe and Mond-Weir type duals are formulated, and various duality relationships between the primal problem and each of the duals are established under invexity and generalized invexity. It is also shown that our results can be viewed as dynamic generalizations of those of the mathematical programming already reported in the literature.     

On a variational problem with non-differentiable constraints

We study the regularity of minimizers and critical points of the Dirichlet energy under an integral constraint given by a non-differentiable function. We obtain existence of a Lipschitz continuous minimizer for a relaxed problem. In two dimensions, some regularity can also be proved for critical points.     

On a class of second order Fuchsian hyperbolic equations

Summary We study a class of second order Fuchsian hyperbolic operators. The well-posedness of the Cauchy problem in a space of regular distributions is proved, together with results on the propagation of singularities of the solution. Moreover we give a representation formula for the distribution solutions of the homogeneous equation.     

On variational problems with obstacles and integral constraints for vector-valued functions

The authors prove existence and regularity for vectorvalued solutions of n-dimensional variational problems with boundary conditions, integral constraints, and obstacles as side conditions. Main emphasis is given to the regularity proof in the case n=2 generalizing a well known technique due to C. B. Morrey. In addition, a regularity result is stated for the general n-dimensional case.Dedicated to Hans Lewy and Charles B. Morrey, Jr.     

First and second order conditions for a class of nondifferentiable optimization problems

This paper gives first and in particular second order necessary and sufficient conditions for a class of nondifferentiable optimization problems in which there are both objective and constraint functions defined in terms of a norm. The conditions are expressed in terms of a Lagrangian function and its derivatives, and use the ideas of feasible directional sequence and subgradients. Certain regularity assumptions are required and for the second order necessary conditions it is shown that the assumption is realistic for polyhedral norms. Illustrative examples are discussed.     

On some second order and fourth order elliptic overdetermined problems

This paper deals with various overdetermined problems. In Sections 2 and 3 we investigate two problems involving the electrostatic potential confined in a region of ³ bounded by two tubular conductors, and we conclude under appropriate assumptions that the only possible configuration of the two conductors is two coaxial circular cylinders. In Section 4 we prove a conjecture of Payne and Schaefer for a fourth order overdetermined problem.     

Optimization with a class of multivariate integral stochastic order constraints

We study convex optimization problems with a class of multivariate integral stochastic order constraints defined in terms of parametrized families of increasing concave functions. We show that utility functions act as the Lagrange multipliers of the stochastic order constraints in this general setting, and that the dual problem is a search over utility functions. Practical implementation issues are discussed.     

On the optimization of a class of second order iterative schemes

This paper analyzes and studies the second order scheme of de Pillis' type, which is used for the solution of a linear system. This study leads to a monoparametric family of second order schemes of the aforementioned type and a method of selecting the optimal one is presented. In addition a number of concluding remarks is made and various applications and examples are given, which effectively show in some cases the superiority of our optimal second order scheme over the fastest first order ones like the SOR and AOR schemes. Many points are also made for the possibility of improving on the convergence rates of the optimal scheme of this paper, which suggest further research in this area.     

Symmetric duality for fractional variational problems with cone constraints

A pair of symmetric fractional variational programming problems is formulated over cones. Weak, strong, converse and self duality theorems are discussed under pseudoinvexity. Static symmetric dual fractional programs are included as special case and corresponding symmetric duality results are merely stated.     

Branching of extremals of some variational problems with phase constraints

Translated from Optimal'nost Upravlyaemykh Dinamicheskikh Sistem, No. 19, pp. 54–62, Vsesoyuznyi Nauchno-Issledovatel'skii Institut Sistemnykh Issledovanii, Moscow (1988).     

Invariant imbedding and a class of variational problems

Consider the minimization of an integral $$I = \int_a^T {[\frac{1}{2}\dot w^2 + F(w,y)]} dy$$ withw(a)=c andw(T)=free. An initial-value problem for the optimizing function is derived directly from the variational problem. It is shown that the solution of the initial-value problem satisfies the usual Euler equation. The Bellman-Hamilton-Jacobi partial differential equation is also treated.     

Existence of solutions to a class of nonlinear second order two-point boundary value problems

In this paper, the existence and multiplicity results of solutions are obtained for the second order two-point boundary value problem −u^″(t)=f(t,u(t)) for all t∈[0,1] subject to u(0)=u^′(1)=0, where f is continuous. The monotone operator theory and critical point theory are employed to discuss this problem, respectively. In argument, quadratic root operator and its properties play an important role.