Generalized partial derivatives and optimality conditions for non-differentiable functions

In [9] a generalization of the concept of a classical gradient was presented, which exists for a broad class of non-differentiable functions. This concept differs from traditional “set-valued”" generalizations by an expansion of a directional derivative into a special orthogonal series. Investigating the coefficients of this series and interpreting them as a kind of partial derivatives formal analogies to properties of classical partial derivatives could be drawn. Moreover the new concept reveals relations between questions appearing within the theory of optimization and results in other branches of mathematics, especially to the theory of harmonic and subharmonic functions, The aim of this paper is to show that with the new tool of generalized partial derivatives necessary and sufficient conditions for optimality can be deduced in the case, that a function under consideration is not differentiable at the point of interest. In such a way criteria of optimality can be formulated using the language of these generalized partial derivatives. Numerical considerations show, that the generalized partial derivatives can (approximately) be calculated without solving integrals by hyperinterpolation on the sphere     

Necessary and sufficient conditions for nonsmooth mathematical programs with equilibrium constraints

In this paper we consider a mathematical program with equilibrium constraints (MPEC) formulated as a mathematical program with complementarity constraints. Then, we derive a necessary optimality result for nonsmooth MPEC on any Asplund space. Also, under generalized convexity assumptions, we establish sufficient optimality conditions for this program in Banach spaces.     

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.     

Necessary and sufficient conditions for discrete and differential inclusions of elliptic type

This paper deals for the first time with the Dirichlet problem for discrete (PD), discrete approximation problem on a uniform grid and differential (PC) inclusions of elliptic type. In the form of Euler-Lagrange inclusion necessary and sufficient conditions for optimality are derived for the problems under consideration on the basis of new concepts of locally adjoint mappings. The results obtained are generalized to the multidimensional case with a second order elliptic operator.     

Necessary and sufficient condition for local minima of a class of nonconvex quadratic programs

The author (1992, 1993) earlier studied the equivalence of a class of 0–1 quadratic programs and their relaxed problems. Thus, a class of combinatorial optimization problems can be solved by solving a class of nonconvex quadratic programs. In this paper, a necessary and sufficient condition for local minima of this class of nonconvex quadratic programs is given; this will be the foundation for study of algorithms.Research supported by Huo Yingdong Educational Foundation '93.     

Necessary and sufficient global optimality conditions for NLP reformulations of linear SDP problems

In this paper we consider the standard linear SDP problem, and its low rank nonlinear programming reformulation, based on a Gramian representation of a positive semidefinite matrix. For this nonconvex quadratic problem with quadratic equality constraints, we give necessary and sufficient conditions of global optimality expressed in terms of the Lagrangian function.     

Linear programming via a quadratic penalty function

We use quadratic penalty functions along with some recent ideas from linearl ₁ estimation to arrive at a new characterization of primal optimal solutions in linear programs. The algorithmic implications of this analysis are studied, and a new, finite penalty algorithm for linear programming is designed. Preliminary computational results are presented.Research supported by grant No. 11-0505 from the Danish Natural Science Research Council SNF.     

Necessary and sufficient conditions for tight equi-difference conflict-avoiding codes of weight three

A conflict-avoiding code (CAC) C of length n and weight k is a collection of k-subsets of such that holds for any , , where . A CAC with maximum code size for given n and k is called optimal. Furthermore, an optimal CAC C is said to be tight equi-difference if holds and any codeword has the form . The concept of a CAC is motivated from applications in multiple-access communication systems. In this paper, we give a necessary and sufficient condition to construct tight equi-difference CACs of weight k = 3 and characterize the code length n’s admitting the condition through a number theoretical approach.      

Necessary and sufficient conditions for the existence of symmetric positive solutions of singular boundary value problems

Necessary and sufficient conditions are obtained for the existence of symmetric positive solutions to the boundary value problem

     

A successive quadratic programming method for a class of constrained nonsmooth optimization problems

In this paper we present an algorithm for solving nonlinear programming problems where the objective function contains a possibly nonsmooth convex term. The algorithm successively solves direction finding subproblems which are quadratic programming problems constructed by exploiting the special feature of the objective function. An exact penalty function is used to determine a step-size, once a search direction thus obtained is judged to yield a sufficient reduction in the penalty function value. The penalty parameter is adjusted to a suitable value automatically. Under appropriate assumptions, the algorithm is shown to produce an approximate optimal solution to the problem with any desirable accuracy in a finite number of iterations.     

Necessary optimality conditions for a special class of bilevel programming problems with unique lower level solution

We consider a bilevel programming problem in Banach spaces whose lower level solution is unique for any choice of the upper level variable. A condition is presented which ensures that the lower level solution mapping is directionally differentiable, and a formula is constructed which can be used to compute this directional derivative. Afterwards, we apply these results in order to obtain first-order necessary optimality conditions for the bilevel programming problem. It is shown that these optimality conditions imply that a certain mathematical program with complementarity constraints in Banach spaces has the optimal solution zero. We state the weak and strong stationarity conditions of this problem as well as corresponding constraint qualifications in order to derive applicable necessary optimality conditions for the original bilevel programming problem. Finally, we use the theory to state new necessary optimality conditions for certain classes of semidefinite bilevel programming problems and present an example in terms of bilevel optimal control.     

A sufficient condition for self-concordance,with application to some classes of structured convex programming problems

Recently a number of papers were written that present low-complexity interior-point methods for different classes of convex programs. The goal of this article is to show that the logarithmic barrier function associated with these programs is self-concordant. Hence the polynomial complexity results for these convex programs can be derived from the theory of Nesterov and Nemirovsky on self-concordant barrier functions. We also show that the approach can be applied to some other known classes of convex programs.This author's research was supported by a research grant from SHELL.On leave from the Eötvös University, Budapest, Hungary. This author's research was partially supported by OTKA No. 2116.     

Second order optimality conditions for a class of control problems governed by non-linear integral equations with application to parabolic boundary control

In the paper necessary and sufficient second order optimality conditions for optimal control problems governed by weakly singular non linear Hammerstein integral equations are derived. They are applied to a semilinear parabolic boundary control problem for the one dimensional heat equation.     

Necessary and sufficient conditions for a curve to be the gradient range of a <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript>-smooth function

We find some necessary and sufficient conditions for a plane curve to be the gradient range of a C ¹-smooth function of two variables. As one of the consequences we give the necessary and sufficient conditions on a continuous function ? under which the differential equation \(\frac{{\partial v}}{{\partial t}} = \varphi \left( {\frac{{\partial v}}{{\partial x}}} \right)\) has nontrivial C ¹-smooth solutions.     

Continuity versus nonexistence for a class of linear stochastic Cauchy problems driven by a Brownian motion

Let A = d/dθ denote the generator of the rotation group in the space C(Γ), where Γ denotes the unit circle. We show that the stochastic Cauchy problem

Convergence of a Reactive Planning Algorithm

The reactive planning algorithms proposed by P. Maes in Artificial Intelligence for real-time decision making and other derived algorithms are empirically known to converge, but no mathematical proof has been proposed. The evolution equations of these incremental algorithms can be translated into discrete time non-linear dynamical systems. We prove the convergence of such dynamical systems, provided the necessary assumptions are fulfilled, and we give an expression for the limit.     

Constant sign and nodal solutions for a class of nonlinear Dirichlet problems

We consider a nonlinear Dirichlet problem with a Carathéodory reaction which has arbitrary growth from below. We show that the problem has at least three nontrivial smooth solutions, two of constant sign and the third nodal. In the semilinear case (i.e., p=2$p=2$), with the reaction f(z,.)$f(z,.)$ being C¹$C^1$ and with subcritical growth, we show that there is a second nodal solution, for a total of four nontrivial smooth solutions. Finally, when the reaction has concave terms and is subcritical and for the nonlinear problem (i.e., 1<p<∞$1<p<\infty$) we show that again we can have the existence of three nontrivial smooth solutions, two of constant sign and a third nodal.     

Multiplicity of positive solutions for a class of quasilinear nonhomogeneous Neumann problems

In this paper we study the existence, nonexistence and multiplicity of positive solutions for nonhomogeneous Neumann boundary value problem of the type

where Ω is a bounded domain in with smooth boundary, 1<p<n,Δ_pu=div(|u|^p-2u) is the p-Laplacian operator, , , (x)0 and λ is a real parameter. The proofs of our main results rely on different methods: lower and upper solutions and variational approach.     

Abelian and Tauberian theorems for a new trigonometric method of summation

We first introduce a new trigonometric method of summation and then prove some Abelian and Tauberian theorems for this method.     

Existence of a nontrivial solution for a class of superquadratic elliptic problems

A nontrivial solution is obtained for a class of superquadratic elliptic problems by variational methods.