Trapping regions for discontinuously coupled systems of evolution variational inequalities and application

We consider initial-boundary value problems for weakly coupled systems of parabolic equations under coupled nonlinear flux boundary condition. Both coupling vector fields and are assumed to be either of competitive or cooperative type, but may otherwise be discontinuous with respect to all their arguments. The main goal is to provide conditions for the vector fields f and g that allow the identification of regions of existence of solutions (so called trapping regions). To this end the problem is transformed to a discontinuously coupled system of evolution variational inequalities. Assuming a generalized outward pointing vector field on the boundary of a rectangle of the dependent variable space, the system of evolution variational inequalities is solved via a fixed point problem for some increasing operator in an appropriate ordered Banach space. The main tools used in the proof are evolution variational inequalities, comparison techniques, and fixed point results in ordered Banach spaces.     

A hinged plate equation and iterated Dirichlet Laplace operator on domains with concave corners

Fourth order hinged plate type problems are usually solved via a system of two second order equations. For smooth domains such an approach can be justified. However, when the domain has a concave corner the bi-Laplace problem with Navier boundary conditions may have two different types of solutions, namely u₁ with and . We will compare these two solutions. A striking difference is that in general only the first solution, obtained by decoupling into a system, preserves positivity, that is, a positive source implies that the solution is positive. The other type of solution is more relevant in the context of the hinged plate. We will also address the higher-dimensional case. Our main analytical tools will be the weighted Sobolev spaces that originate from Kondratiev. In two dimensions we will show an alternative that uses conformal transformation. Next to rigorous proofs the results are illustrated by some numerical experiments for planar domains.     

An upper bound for SLP using first and total second moments

In 1987, J. Dulá considered the problem of finding an upper bound for the expectation of a so-called simplicial function of a random vector and used for this purpose first and total second moments. Under the same moment conditions we consider some different cases of recourse functions and demonstrate how the related moment problems can be solved by solving nonsmooth (unconstrained) optimization problems and thereafter satisfying simple linear constraint systems.     

Hybrid meta-heuristics with VNS and exact methods: alication to large unconditional and conditional vertex $$$$-centre roblems

Large-scale unconditional and conditional vertex \(p\)-centre problems are solved using two meta-heuristics. One is based on a three-stage approach whereas the other relies on a guided multi-start principle. Both methods incorporate Variable Neighbourhood Search, exact method, and aggregation techniques. The methods are assessed on the TSP dataset which consist of up to 71,009 demand points with \(p\) varying from 5 to 100. To the best of our knowledge, these are the largest instances solved for unconditional and conditional vertex \(p\)-centre problems. The two proposed meta-heuristics yield competitive results for both classes of problems.     

Designing a regression experiment in Hilbert space

Two forms of a priori information about unknown parameters in observing linear regression with noise in a Hilbert space are considered. In each section one gets best restricted linear unbiased estimates of the unknown parameters generalizing known results about the finite-dimensional case. Problems of choice of optimal designs are solved for bounded closed sets which are natural from the point of view of applications. The results are compared with those which hold without a priori information.Translated from Teoriya Sluchainykh Protsessov, No. 16, pp. 23–28, 1988.     

Large-scale semidefinite programming via a saddle point Mirror-Prox algorithm

In this paper, we first demonstrate that positive semidefiniteness of a large well-structured sparse symmetric matrix can be represented via positive semidefiniteness of a bunch of smaller matrices linked, in a linear fashion, to the matrix. We derive also the “dual counterpart” of the outlined representation, which expresses the possibility of positive semidefinite completion of a well-structured partially defined symmetric matrix in terms of positive semidefiniteness of a specific bunch of fully defined submatrices of the matrix. Using the representations, we then reformulate well-structured large-scale semidefinite problems into smooth convex–concave saddle point problems, which can be solved by a Prox-method developed in [6] with efficiency . Implementations and some numerical results for large-scale Lovász capacity and MAXCUT problems are finally presented.      

Adaptive boundary control of stochastic linear distributed parameter systems described by analytic semigroups

A stochastic adaptive control problem is formulated and solved for some unknown linear, stochastic distributed parameter systems that are described by analytic semigroups. The control occurs on the boundary. The highest-order operator is assumed to be known but the lower-order operators contain unknown parameters. Furthermore, the linear operators of the state and the control on the boundary contain unknown parameters. The noise in the system is a cylindrical white Gaussian noise. The performance measure is an ergodic, quadratic cost functional. For the identification of the unknown parameters a diminishing excitation is used that has no effect on the ergodic cost functional but ensures sufficient excitation for strong consistency. The adaptive control is the certainty equivalence control for the ergodic, quadratic cost functional with switchings to the zero control.This research was partially supported by NSF Grants ECS-9102714, ECS-9113029, and DMS-9305936.     

Use of chance-constrained programming to account for stochastic variation in theA-matrix of large-scale linear programs: A forestry application

Linear programming (LP) is widely used to select the manner in which forest lands are managed. Because of the nature of forestry, this application has several unique characteristics. For example, the models consider many different management actions that take place over many years, thus resulting in very large LP formulations with diverse data. In addition, almost none of the data are known with certainty. The most pervasive occurrence of stochastic information is in the production coefficients, which indicate the uncertain response of the managed forest ecosystem to various management options. A chance-constrained approach to handling this uncertainty would often be appropriate in forestry applications —managers and decision makers would like to specify a probability with which uncertain constraints are met. Unfortunately, chance-constrained procedures forA-matrix uncertainty produce nonlinear programming problems, which cannot currently be solved for large-scale forestry applications. This paper utilizes a Monte Carlo simulation approach (a linear program is repeatedly solved with randomly perturbedA-matrix coefficients) to describe the distribution of total output when the individual production coefficients are random. An iterative procedure for chance-constraining feasibility is developed and demonstrated with this sort of randomA-matrix. An iterative approach is required because the mean and variance of total output are unknown functions of the randomA-matrix coefficients and the level of output required. This approach may have applications in other fields as well.     

Eine Methode zur Berechnung quasistationärer Temperaturzustände bei gesteuerten Wärmeleitvorgängen in festen Körpern

Summary In a recent paperVodika solves a special problem of heat conduction with local and time-dependent boundary conditions. As the method does not seem very suitable for solving problems of this kind, the solution for the most general case is given. As an example the problem treated byVodika is solved, and the equality of both results is shown.     

Newton’s Method and Frobenius–Dieudonné Theorem in Nonnormable Spaces

We establish, via auxiliary categories, a version of Newtons method applicable to nonnormable spaces. We illustrate it with an application to perturbation of functional equations on unbounded domains. Under suitable conditions, the known solution of the unperturbed equation initiates a sequence of Newton iterates that converges to the solution of the perturbed equation. The conditions posed are those required to make Newtons method work. We also generalize the Frobenius–Dieudonné existence-uniqueness theorem for initial value problems for a vector variable.     

Diffraction of the Plane Wave on a Periodic Grating Composed of Thin Chiral Half-Planes

The diffraction of the normally incident plane wave by a grating consisting of thin semi-infinite chiral slabs is considered. The chiral slabs are simulated by appropriate transition boundary conditions. The problem is simplified by decoupling the and components with the help of a similarity transformation. Then the problem is reduced to scalar Riemann–Hilbert problems and is solved in explicit form. An expansion of the diffracted field in plane waves is obtained, and numerical results are discussed. Bibliography: 9 titles.     

Numerical solution of minimum norm problems of distributed control of vibrations

This paper is concerned with the distributed control of a vibration process that can be described by a differential equation for a Hilbert-spacevalued functiony: [0, ) H. The control functions on the right-hand side of this equation are taken fromL ([0, ),H) equipped with the essential supremum norm. To be solved is the problem of time-minimal null-controllability by norm-bounded controls. This problem is essentially reduced to solving the problem of minimum norm control on a given time interval. This is solved via its dual problem which is approximately solved by truncation and discretization. Numerical results are presented for a vibrating string and a vibrating beam.     

Approximate <Emphasis Type="Italic">k</Emphasis>-MSTs and <Emphasis Type="Italic">k</Emphasis>-Steiner trees via the primal-dual method and Lagrangean relaxation

Garg [10] gives two approximation algorithms for the minimum-cost tree spanning k vertices in an undirected graph. Recently Jain and Vazirani [15] discovered primal-dual approximation algorithms for the metric uncapacitated facility location and k-median problems. In this paper we show how Gargs algorithms can be explained simply with ideas introduced by Jain and Vazirani, in particular via a Lagrangean relaxation technique together with the primal-dual method for approximation algorithms. We also derive a constant factor approximation algorithm for the k-Steiner tree problem using these ideas, and point out the common features of these problems that allow them to be solved with similar techniques.     

A New Lagrangian Relaxation Based Algorithm for a Class of Multidimensional Assignment Problems

Large classes of data association problems in multiple targettracking applications involving both multiple and single sensorsystems can be formulated as multidimensional assignment problems.These NP-hard problems are large scale and sparse with noisyobjective function values, but must be solved inreal-time. Lagrangian relaxation methods have proven to beparticularly effective in solving these problems to the noise levelin real-time, especially for dense scenarios and for multiple scansof data from multiple sensors. This work presents a new class ofconstructive Lagrangian relaxation algorithms that circumvent some ofthe deficiencies of previous methods. The results of severalnumerical studies demonstrate the efficiency and effectiveness of thenew algorithm class.     

Boundary value problems for some nonlinear systems with singular \phi-laplacian

Systems of differential equations of the form

Balanced Incomplete Block Designs with Block Size 7

The object of this paper is the construction of balanced incomplete block designs with k=7. This paper continues the work begun by Hanani, who solved the construction problem for designs with a block size of 7, and with =6, 7, 21 and 42. The construction problem is solved here for designs with > 2 except for v=253, = 4,5 ; also for = 2, the number of unconstructed designs is reduced to 9 (1 nonexistent, 8 unknown).     

Hereditary optimal control problems: Numerical method based upon a padé approximation

In this paper, we consider a particular approximation scheme which can be used to solve hereditary optimal control problems. These problems are characterized by variables with a time-delayed argumentx(t – ). In our approximation scheme, we first replace the variable with an augmented statey(t) x(t - ). The two-sided Laplace transform ofy(t) is a product of the Laplace transform ofx(t) and an exponential factor. This factor is approximated by a first-order Padé approximation, and a differential relation fory(t) can be found. The transformed problem, without any time-delayed argument, can then be solved using a gradient algorithm in the usual way. Four problems are solved to illustrate the validity and usefulness of this technique.This research was supported in part by the National Aeronautics and Space Administration under NASA Grant NCC-2-106.     

Solving structured nonsmooth convex optimization with complexity $$\mathcal {O}(\varepsilon ^{-1/2})$$

This paper describes an algorithm for solving structured nonsmooth convex optimization problems using the optimal subgradient algorithm (OSGA), which is a first-order method with the complexity \(\mathcal {O}(\varepsilon ^{-2})\) for Lipschitz continuous nonsmooth problems and \(\mathcal {O}(\varepsilon ^{-1/2})\) for smooth problems with Lipschitz continuous gradient. If the nonsmoothness of the problem is manifested in a structured way, we reformulate the problem so that it can be solved efficiently by a new setup of OSGA (called OSGA-V) with the complexity \(\mathcal {O}(\varepsilon ^{-1/2})\). Further, to solve the reformulated problem, we equip OSGA-O with an appropriate prox-function for which the OSGA-O subproblem can be solved either in a closed form or by a simple iterative scheme, which decreases the computational cost of applying the algorithm for large-scale problems. We show that applying the new scheme is feasible for many problems arising in applications. Some numerical results are reported confirming the theoretical foundations.     

Using copositivity for global optimality criteria in concave quadratic programming problems

In this note we specify a necessary and sufficient condition for global optimality in concave quadratic minimization problems. Using this condition, it follows that, from the perspective of worst-case complexity of concave quadratic problems, the difference between local and global optimality conditions is not as large as in general. As an essential ingredient, we here use the-subdifferential calculus via an approach of Hiriart-Urruty and Lemarechal (1990).