Clark's Theorem Research Articles

Exact Penalization, Level Function Method, and Modified Cutting-Plane Method for Stochastic Programs with Second Order Stochastic Dominance Constraints

Level function methods and cutting-plane methods have been recently proposed to solve stochastic programs with stochastic second order dominance (SSD) constraints. A level function method requires an exact penalization setup because it can only be applied to the objective function, not the constraints. Slater constraint qualification (SCQ) is often needed for deriving exact penalization. It is well known that SSD usually does not satisfy SCQ and various relaxation schemes have been proposed so that the relaxed problem satisfies the SCQ. In this paper, we show that under some moderate conditions the desired constraint qualification can be guaranteed through some appropriate reformulation of the constraints rather than relaxation. Exact penalization schemes based on $L_1$-norm and $L_\infty$-norm are subsequently derived through Robinson's error bound on convex systems and Clarke's exact penalty function theorem. Moreover, we propose a modified cutting-plane method which constructs a cutting-plane through t...

Multiple Solutions for a Fractional Difference Boundary Value Problem via Variational Approach

By establishing the corresponding variational framework and using the mountain pass theorem, linking theorem, and Clark theorem in critical point theory, we give the existence of multiple solutions for a fractional difference boundary value problem with parameter. Under some suitable assumptions, we obtain some results which ensure the existence of a well precise interval of parameter for which the problem admits multiple solutions. Some examples are presented to illustrate the main results.

Multiple solutions for a fourth-order difference boundary value problem with parameter via variational approach

By establishing the corresponding variational framework, and using the mountain pass theorem, linking theorem and Clark theorem in critical point theory, we give the existence of multiple solutions for a fourth-order difference boundary value problem with parameter. Under some suitable assumptions we obtain some results which ensure the existence of well precise interval of parameter for which the problem admits multiple solutions. Some examples are presented to illustrate the main results.

Existence and multiplicity of homoclinic solutions for second-order discrete Hamiltonian systems with subquadratic potential

In the present paper, we deal with the existence and multiplicity of homoclinic solutions of the second-order self-adjoint discrete Hamiltonian system where and are neither autonomous nor periodic in n. Under the assumption that is indefinite sign and subquadratic as and p(n) and L(n) are N × N real symmetric positive definite matrices for all , we establish some existence criteria to guarantee that the above system has at least one or multiple homoclinic solutions by using Clark's theorem in critical point theory.

Multiple periodic solutions for non-linear difference systems involving the p-Laplacian

Using the three critical points theorem, Clark's theorem and the Morse theory, multiple periodic solutions for non-linear difference systems involving the p-Laplacian are obtained by variational methods.

Existence and Multiplicity of Nontrivial odd Periodic Solutions of Second order Hamiltonian Systems with Bi-even Subquadratic Potentials

Existence and multiplicity of nontrivial odd periodic solutions of second order Hamiltonian systems(x|¨)(t) + V_x(t,x(t)) = 0 with bi-even subquadratic potentials are discussed by Clark Theorem in variational methods.

On a non-smooth eigenvalue problem in Orlicz–Sobolev spaces

This article studies a non-smooth eigenvalue problem for a Dirichlet boundary value inclusion on a bounded domain Ω which involves a φ-Laplacian and the generalized gradient in the sense of Clarke of a locally Lipschitz function depending also on the points in Ω. Specifically, the existence of a sequence of eigensolutions satisfying in addition certain asymptotic and locational properties is established. The approach relies on an approximation process in a suitable Orlicz–Sobolev space by eigenvalue problems in finite-dimensional spaces for which one can apply a finite-dimensional, non-smooth version of the Ljusternik–Schnirelman theorem. As a byproduct of our analysis, a version of Aubin–Clarke's theorem in Orlicz spaces is obtained.

A remark on multiple solutions for a nonlinear eigenvalue problem

Multiple solutions are shown to exist for the system Bu = λF(u) , where λ > 0 , B is a positive definite matrix and F is ‘superquadratic’ or ‘subquadratic’. We make use of Clark's theorem in critical point theory to obtain our results and provide examples to show that the results are sharp.

Unitary Space–Time Constellation Design Based on the Chernoff Bound of the Pairwise Error Probability

Unitary space-time constellation design is considered for noncoherent multiple-antenna communications, where neither the transmitter nor the receiver knows the fading coefficients of the channel. By employing the Clarke's subdifferential theorem of the sum of the kappa largest singular values of a unitary matrix, we present a numerical optimization procedure for finding unitary space-time signal constellations of any dimension. The Chernoff bound of the pairwise error probability is used directly as a design criterion. The constellations are found by performing gradient descent search on a family ldquosurrogaterdquo functions that converge to the maximum pairwise error probability. The complexity of the search procedure increases with the dimension and the size of the constellation, but it can be considered to be acceptable for an off-line design procedure. Since the designed constellations are unstructured, and, thus, require an exhaustive search over all codewords in decoding, their main practical value is to serve as constellation design performance benchmarks. We compare the performance of the new constellations to that of some other well-known constellations. Computer simulation results illustrate typically about 0.4-3.5 dB performance gains.

On the solvability of a boundary value problem for a fourth-order ordinary differential equation

We study the existence and multiplicity of nontrivial periodic solutions for a semilinear fourth-order ordinary differential equation arising in the study of spatial patterns for bistable systems. Variational tools such as the Brezis–Nirenberg theorem and Clark theorem are used in the proofs of the main results.

Periodic and homoclinic solutions of some semilinear sixth-order differential equations

In this paper we study the existence of periodic solutions of the sixth-order equation u vi+Au iv+Bu″+u−u 3=0, where the positive constants A and B satisfy the inequality A 2<4 B. The boundary value problem (P) is considered with the boundary conditions u(0)=u″(0)=u iv(0)=0,u(L)=u″(L)=u iv(L)=0. Existence of nontrivial solutions for (P) is proved using a minimization theorem and a multiplicity result using Clark's theorem. We study also the homoclinic solutions for the sixth-order equation u vi+Au iv+Bu″−u+a(x)u|u| σ=0, where a is a positive periodic function and σ is a positive constant. The mountain-pass theorem of Brezis–Nirenberg and concentration-compactness arguments are used.

Periodic and Homoclinic Solutions of Extended Fisher–Kolmogorov Equations

In this paper we study the existence of periodic solutions of the fourth-order equations uiv−pu″−a(x)u+b(x)u3=0 and uiv−pu″+a(x)u−b(x)u3=0, where p is a positive constant, and a(x) and b(x) are continuous positive 2L-periodic functions. The boundary value problems (P1) and (P2) for these equations are considered respectively with the boundary conditions u(0)=u(L)=u″(0)=u″(L)=0. Existence of nontrivial solutions for (P1) is proved using a minimization theorem and a multiplicity result using Clark's theorem. Existence of nontrivial solutions for (P2) is proved using the symmetric mountain-pass theorem. We study also the homoclinic solutions for the fourth-order equation uiv+pu″+a(x)u−b(x)u2−c(x)u3=0, where p is a constant, and a(x), b(x), and c(x) are periodic functions. The mountain-pass theorem of Brezis and Nirenberg and concentration-compactness arguments are used.

Universal Higher Order Bernoulli Numbers and Kummer and Related Congruences

We define higher or arbitrary order universal Bernoulli numbers and higher order universal Bernoulli–Hurwitz numbers. We deduce a universal first-order Kummer congruence and a congruence for the higher order universal Bernoulli–Hurwitz numbers from Clarke's universal von Staudt theorem. We also establish other Kummer-type congruences for the higher order universal Bernoulli numbers and for the universal Nörlund polynomials, generalizing the author's previous work.

Clar Polynomials of Large Benzenoid Systems

We discuss the constructions of Clar-like valence structures and the associated Clar polynomials for large benzenoids. We have shown that Clar polynomial satisfy a theorem analogous to the Clarke's theorem for the characteristic polynomial of graphs, which allows one to reconstruct the polynomial of a system from the polynomials associated with qualified substructures. In the case of Clar's polynomial the substructures are obtained by deleting individual benzene rings from the considered benzenoid, one at a time. This property of Clar polynomial allows one to construct the polynomial of larger systems from known Clar's polynomials of smaller benzenoids.

Convergence of stochastic algorithms: from the Kushner–Clark theorem to the Lyapounov functional method

In the first part of this paper a global Kushner–Clark theorem about the convergence of stochastic algorithms is proved: we show that, under some natural assumptions, one can ‘read' from the trajectories of its ODE whether or not an algorithm converges. The classical stochastic optimization results are included in this theorem. In the second part, the above smoothness assumption on the mean vector field of the algorithm is relaxed using a new approach based on a path-dependent Lyapounov functional. Several applications, for non-smooth mean vector fields and/or bounded Lyapounov function settings, are derived. Examples and simulations are provided that illustrate and enlighten the field of application of the theoretical results.

Convergence of stochastic algorithms: from the Kushner–Clark theorem to the Lyapounov functional method

In the first part of this paper a global Kushner–Clark theorem about the convergence of stochastic algorithms is proved: we show that, under some natural assumptions, one can ‘read' from the trajectories of its ODE whether or not an algorithm converges. The classical stochastic optimization results are included in this theorem. In the second part, the above smoothness assumption on the mean vector field of the algorithm is relaxed using a new approach based on a path-dependent Lyapounov functional. Several applications, for non-smooth mean vector fields and/or bounded Lyapounov function settings, are derived. Examples and simulations are provided that illustrate and enlighten the field of application of the theoretical results.

Logic Programming from the Perspective of Algebraic Semantics

We present an approach to foundations of logic programming in which the connection with algebraic semantics becomes apparent. The approach is based on omega-Herbrand models instead of conventional Herbrand models. We give a proof of Clark's theorem on completeness of SLD-resolution by methods of the algebraic semantics. We prove the existence property for definite programs.

Some results on unstable polynomial algebras over the steenrod algebra

In this paper we give a condition under which an unstable polynomialA p -algebra can be obtained from the modular polynomial algebra. From this we car deduce both the Aguades theorem[2] and Clark's theorem[4].

Clark's theorem for semi-infinite convex programs

In 1961, Clark proved that if either the feasible region of a linear program or its dual is nonempty and bounded, then the other is unbounded. Recently, Duffin has extended this result to a convex program and its Lagrangian dual. Moreover, Duffin showed that under this boundedness assumption there is no duality gap. The purpose of this paper is to extend Duffin's results to semi-infinite programs.

Clark's Theorem on linear programs holds for convex programs

Given a linear minimization program, then there is an associated linear maximization program termed the dual. F. E. Clark proved the following theorem. "If the set of feasible points of one program is bounded, then the set of feasible points of the other program is unbounded." A convex program is the minimization of a convex function subject to the constraint that a number of other convex functions be nonpositive. As is well known, a dual maximization problem can be defined in terms of the Lagrange function. The dual objection function is the infimum of the Lagrange function. The feasible Lagrange multipliers are those satisfying: (i) the multipliers are nonnegative and (ii) the dual objective function is not negative infinity. It is found that Clark's Theorem applies unchanged to dual convex programs. Moreover, the programs have equal values.