On conditions to have bounded multipliers in locally lipschitz programming

For a nonlinear programming problem with locally Lipschitz objective and inequality constraint functions and continuously differentiable equality constraint functions, a necessary and sufficient condition is presented for the set of multiplier vectors to be nonempty and bounded.This author's work was supported by the National Science Foundation under Grant No. MCS 78-06716.     

Reduction from a Semi-Infinite Interval to a Finite Interval of a Nonlinear Boundary Value Problem for a System of Second-Order Equations with a Small Parameter

We consider a boundary value problem over a semi-infinite interval for a nonlinear autonomous system of second-order ordinary differential equations with a small parameter at the leading derivatives. We impose certain constraints on the Jacobian under which a solution to the problem exists and is unique. To transfer the boundary condition from infinity, we use the well-known approach that rests on distinguishing the variety of solutions satisfying the limit condition at infinity. To solve an auxiliary Cauchy problem, we apply expansions of a solution in the parameter.     

A convergence theorem of Rosen’s gradient projection method

Since Rosen’s gradient projection method was published in 1960, a rigorous convergence proof of his method has remained an open question. A convergence theorem is given in this paper. Part of this author’s work was done while he studied at the Department of Mathematics, University of California at Santa Barbara, and was supported by the National Science Foundation under Grant No. MCS83-14977. Part of this author’s work was done while he visited the Computer Science Department, University of Minnesota, Minneapolis, and was supported by the National Science Foundation under Grant No. MCS81-01214.     

Stochastic variational formula for fundamental solutions of parabolic PDE

Letp(t, x, y) be the fundamental solution of a linear, second order partial differential equation of parabolic type. The functionI = –logp satisfies a nonlinear parabolic equation, which is the dynamic programming equation associated with a control problem of stochastic calculus of variations type. This gives a stochastic variational formula forp. The proof depends on a result of Molchanov about the asymptotic behavior ofp(t, x, y) for smallt. This research was partially supported by NSF under Grant No. MCS 8121940 and by AFOSR under Grant No. AF-AFOSR-81-0116-C.     

Necessary conditions and a sufficient condition for the Fischer-Hadamard inequalities

It is well known that a matrix, all of whose principal minors are positive, satisfies the Fischer-Hadamard inequalities if and only if it is weakly sign symmetric. In this paper we consider the general case of matrices whose principal minors may be nonpositive. Necessary conditions and a sufficient condition for the Fischer-Hadamard inequalities to hold are given in the general case.     

Optimal switching in coplanar orbit transfer

The set of all controls that satisfy the Weierstrass necessary condition for optimality in the problem of time-open, coplanar orbit transfer via impulses is presented, along with the switching relations that must be satisfied at a corner in an optimal trajectory. This includes detailed data for eccentricities near unity. This study takes advantage of recently discovered closed-form solutions for the switching surfaces of this problem.Portions of this work were supported by NASA Contract No. NASr-54(06) and by NASA Grant No. NGR-06-003-033.     

Positivity and stability for some partial functional differential equations

The aim of this work is to study the stability for some linear partial functional differential equations. We assume that the linear part is non-densely defined and satisfies the Hille-Yosida condition. Using the positiveness, we give nessecary and sufficient conditions independently of the delay to ensure the uniform exponential stability of the solution semigroup. An application is given for a reaction diffusion equation with several delays. RID="h1" ID="h1"This work is supported by the Moroccan Grant PARS MI 36 and TWAS Grant under contract: No. 00-412 RG/MATHS/AF/AC.     

Proper inequality constraints and maximization of index vectors

For obtaining the set of all quasi-supremal index vectors (or all maximal index vectors, or all Pareto-optimal solutions) of a multiple-objective optimization problem, we present, in this paper, the method of proper inequality constraints, which does not rely on any convexity condition at all, but by which one can obtain the entire desired set. This method is based on the observation that optimizing the index of one of the objectives, with some arbitrary bounds assigned to all other objectives, may still result in inferior solutions, unless these bounds areproper. Various necessary and/or sufficient conditions are presented for the properness test.This work was supported by the National Science Foundation under Grant No. GK-32701.     

Error bounds for monotone linear complementarity problems

We give a bound on the distance between an arbitrary point and the solution set of a monotone linear complementarity problem in terms of a condition constant that depends on the problem data only and a residual function of the violations of the complementary problem conditions by the point considered. When the point satisfies the linear inequalities of the complementarity problem, the residual consists of the complementarity condition plus its square root. This latter term is essential and without it the error bound cannot hold. We also show that another natural residual that has been employed to bound errors for strictly monotone linear complementarity problems fails to bound errors for the monotone case considered here. Sponsored by the United States Army under contract No. DAAG29-80-C-0041. This material is based on research sponsored by National Foundation Grant DCR-8420963 and Air Force Office of Scientific Research Grant AFOSR-ISSA-85-00080.     

On the number of solutions to a class of linear complementarity problems

In this note, we consider the linear complementarity problemw = Mz + q, w 0, z 0, w ^T z = 0, when all principal minors ofM are negative. We show that for such a problem for anyq, there are either 0, 1, 2, or 3 solutions. Also, a set of sufficiency conditions for uniqueness is stated.The work of both authors is partially supported by a grant from the National Science Foundation, MCS 77-03472.     

Maximal vectors and multi-objective optimization

Maximal vector andweak-maximal vector are the two basic notions underlying the various broader definitions (like efficiency, admissibility, vector maximum, noninferiority, Pareto's optimum, etc.) for optimal solutions of multi-objective optimization problems. Moreover, the understanding and characterization of maximal and weak-maximal vectors on the space of index vectors (vectors of values of the multiple objective functions) is fundamental and useful to the understanding and characterization of Pareto-optimal and weak-optimal solutions on the space of solutions.This paper is concerned with various characterizations of maximal and weak-maximal vectors in a general subset of the EuclideanN-space, and with necessary conditions for Pareto-optimal and weak-optimal solutions to a generalN-objective optimization problem having inequality, equality, and open-set constraints on then-space. A geometric method is described; the validity of scalarization by linear combination is studied, and weak conditioning by directional convexity is considered; local properties and a fundamental necessary condition are given. A necessary and sufficient condition for maximal vectors in a simplex or a polyhedral cone is derived. Necessary conditions for Pareto-optimal and weak-optimal solutions are given in terms of Lagrange multipliers, linearly independent gradients, Jacobian and Gramian matrices, and Jacobian determinants.Several advantages in approaching the multi-objective optimization problem in two steps (investigate optimal index vectors on the space of index vectors first, and study optimal solutions on the specific space of solutions next) are demonstrated in this paper.This work was supported by the National Science Foundation under Grant No. GK-32701.     

A constructive characterization ofQ o-matrices with nonnegative principal minors

In [2] we characterized the class of matrices with nonnegative principla minors for which the linear-complementarity problem always has a solution. That class is contained in the one we study here. Our main result gives a finitely testable set of necessary and sufficient conditions under which a matrix with nonnegative principal minors has the property that if a corresponding linear complementarity problem is feasible then it is solvable. In short, we constructively characterize the matrix class known asQ _o ∩P _o . Research partially supported by the National Science Foundation Grant DMS-8420623 and U.S. Department of Energy Contract DE-AA03-76SF00326, PA # DE-AS03-76ER72018.     

Finding all solutions to polynomial systems and other systems of equations

In a previous paper, the authors suggested a procedure for obtaining all solutions to certain systems ofn equations inn complex variables. The idea was to start with a trivial system of equations to which all solutions were easily known. The trivial system was then perturbed into the given system. During the perturbation process, one followed the solution paths from each of the trivial solutions into the solutions of the given system. All solutions to the given system were thereby obtained.This paper utilizes a different approach that eliminates the requirement of the previous paper for a leading dominating term in each equation. We add a dominating term artificially and then fade it. Also we rely on mathematically more fundamental concepts from differential topology. These advancements permit the calculation of all solutions to arbitrary polynomials and to various other systems ofn equations inn complex variables. In addition, information on the number of solutions can be obtained without calculation.Work supported in part by NSF Grant No. MCS77-15509 and ARO Grant No. DAAG-29-78-G-0160.Work supported in part by ARO Grant No. DAAG-29-78-G-0160     

Duality in infinite dimensional linear programming

We consider the class of linear programs with infinitely many variables and constraints having the property that every constraint contains at most finitely many variables while every variable appears in at most finitely many constraints. Examples include production planning and equipment replacement over an infinite horizon. We form the natural dual linear programming problem and prove strong duality under a transversality condition that dual prices are asymptotically zero. That is, we show, under this transversality condition, that optimal solutions are attained in both primal and dual problems and their optimal values are equal. The transversality condition, and hence strong duality, is established for an infinite horizon production planning problem.This material is based on work supported by the National Science Foundation under Grant No. ECS-8700836.     

Solution differentiability and continuation of Newton's method for variational inequality problems over polyhedral sets

In this paper, we derive some further differentiability properties of solutions to a parametric variational inequality problem defined over a polyhedral set. We discuss how these results can be used to establish the feasibility of continuation of Newton's method for solving the variational problem in question.This work was based on research supported by the National Science Foundation under Grant No. ECS-87-17968.     

Asymptotic behavior of a system of interacting nuclear-space-valued stochastic differential equations driven by Poisson random measures

In this paper we study a system of interacting stochastic differential equations taking values in duals of nuclear spaces driven by Poisson random measures. We also consider the McKean-Vlasov equation associated with the system. We show that under suitable conditions the system has a unique solution and the sequence of its empirical distributions converges to the solution of the McKean-Vlasov equation when the size of the system tends to infinity. The results are applied to the voltage potentials of a large system of neurons and the limiting distribution of the empirical measure is obtained.This research was supported by the National Science Foundation, the Air Force Office of Scientific Research under Grant No. F49620-92-J-0154, and the Army Research Office under Grant No DAAL03-92-G-0008.     

Blow-up and existence for fast diffusion equations with general nonlinearities

1.IntroductionInthispaperweconsidertheCauchyproblemforthefastdiffusionequationwheremaxandpositivefunction.Thistypeofequationhasbeenextensivelystudiedasamathematicalmodelofalotofphysicalproblems(see[1-3]).Amajortopicofstudyistheexistenceandnonexistenc...     

An extension of some optimal properties of principal components

Summary Typically, optimal properties for principal components concern the simultaneous minimization of eigenvalues of certain covariance matrices which measure the goodness of an approximation. Many population criteria like total variance and generalized variance, which are increasing functions of the eigenvalues, are then minimized by the best approximator. In other situations, the criterion may not be a monotone function of the eigenvalues. In Theorem 1, we derive a general optimal class based on the non-negative definite ordering of covariance matrices. Sponsored by the United States Army under Contract No. DAAG29-80-C-0041 and the National Science Foundation under Grant No. MCS77-09574.     

A decomposition of time scales for iterative computation of time-optimal controls

An iterative procedure, exploiting the multiple time scale properties of systems with slow and fast modes, is presented. This procedure solves the time-optimal problem for higher-order systems iteratively via the solutions of simple second-order problems. Computational complexity is so reduced that fourth-order problems can be solved using a microprocessor or programmable calculator.This work was supported in part by the Joint Services Electronics Program (US Army, US Navy, and US Air Force) under Contract No. DAAB-07-72-C-0259, in part by the US Air Force under Grant No. AFOSR-73-2570, and in part by the National Science Foundation under Grant No. ENG 74-20091.     

A class of singularly perturbed,nonlinear, fixed-endpoint control problems

Singular perturbation techniques are applied to a class of nonlinear, fixed-endpoint control problems to decompose the full-order problem into three lower-order problems, namely, the reduced problem and the left and right boundary-layer problems. The boundary-layer problems are linear-quadratic and, contrary to previous singular perturbation works, the reduced problem has a simple formulation. The solutions of these lower-order problems are combined to yield an approximate solution to the full nonlinear problem. Based on the properties of the lower-order problems, the full problem is shown to possess an asymptotic series solution.This work was supported in part by the National Science Foundation under Grant No. ENG-47-20091 and in part by the US Air Force under Grant No. AFOSR-73-2570.The author acknowledges the helpful suggestions of Professor P. V. Kokotovic, University of Illinois, Urbana, Illinois.