Obstacle-type problems for minimal surfaces

We study certain obstacle-type problems involving standard and nonlocal minimal surfaces. We obtain optimal regularity of the solution and a characterization of the free boundary.     

A geometric Cauchy problem for timelike minimal surfaces

We investigate two-dimensional timelike surfaces in a Lorentz manifold (X,g). It is shown that orientable surfaces with two spacelike boundary components (homeomorphic toS ¹) are necessarily of topological type [0,1] x S¹. We treat the initial value problem of a string (known from physics) as a purely geometric problem: Find a minimal surface which is specified by an initial curve and by a distribution of timelike tangent planes along . We prove the local existence and uniqueness of and also obtain global existence for special types (X,g). Global existence does not generally hold; we give a counter-example, which can be interpreted as a string collapsing into a black hole.     

A TRUST REGION ALGORITHM VIA BILEVEL LINEAR PROGRAMMING FOR SOLVING THE GENERAL MULTICOMMODITY MINIMAL COST FLOW PROBLEMS

This paper proposes a nonmonotonic backtracking trust region algorithm via bilevel linear programming for solving the general multicommodity minimal cost flow problems. Using the duality theory of the linear programming and convex theory, the generalized directional derivative of the general multicommodity minimal cost flow problems is derived. The global convergence and superlinear convergence rate of the proposed algorithm are established under some mild conditions.     

A method for solving stochastic eigenvalue problems

We present a novel approach for calculating stochastic eigenvalues of differential and integral equations as well as for random matrices. Five examples based on very different types of problem have been analysed and detailed numerical results obtained. It would seem that the method has considerable promise. The essence of the method is to replace the stochastic eigenvalue problem λ(ξ)?(ξ)=A(ξ)?(ξ), where ξ is a set of random variables, by the introduction of an auxiliary equation in which . This changes the problem from an eigenvalue one to an initial value problem in the new pseudo-time variable t. The new linear time-dependent equation may then be solved by a polynomial chaos expansion (PCE) and the stochastic eigenvalue and its moments recovered by a limiting process. This technique has the advantage of avoiding the non-linear terms in the conventional method of stochastic eigenvalue calculation by PCE, but it does introduce an additional, ‘pseudo-time’, independent variable t. The paper illustrates the viability of this approach by application to several examples based on realistic problems.     

A martingale approach to minimal surfaces

We provide a probabilistic approach to studying minimal surfaces in R³. After a discussion of the basic relationship between Brownian motion on a surface and minimality of the surface, we introduce a way of coupling Brownian motions on two minimal surfaces. This coupling is then used to study two classes of results in minimal surface theory, maximum principle-type results, such as weak and strong halfspace theorems and the maximum principle at infinity, and Liouville theorems.     

A Weierstrass type representation for minimal surfaces in Sol

The normal Gauss map of a minimal surface in the model space of solvegeometry is a harmonic map with respect to a certain singular Riemannian metric on the extended complex plane.

     

A note on solving large p-median problems

In a previous paper we presented a tree search algorithm for the p-median problem, the problem of locating p facilities (medians) on a network, which was based upon La grangean relaxation and subgradient optimisation. That algorithm solved (optimally) problems with an arbitrary number of medians and having up to 200 vertices.In this note we show that it is possible to enhance that algorithm to solve (optimally) problems having up to 900 vertices using the Cray-1S computer.     

A heuristic search based on diversity for solving combinatorial problems

Journal of Heuristics - In this paper we propose a novel heuristic search for solving combinatorial optimization problems which we call Diverse Search (DS). Like beam search, this constructive...     

Some Picard theorems for minimal surfaces

This paper deals with the study of those closed subsets for which the following statement holds:

If is a properly immersed minimal surface in of finite topology that is eventually disjoint from then has finite total curvature.

The same question is also considered when the conclusion is finite type or parabolicity.

     

A general method for solving constrained optimization problems

In this paper a general problem of constrained minimization is studied. The minima are determined by searching for the asymptotical values of the solutions of a suitable system of ordinary differential equations.For this system, if the initial point is feasible, then any trajectory is always inside the set of constraints and tends towards a set of critical points. Each critical point that is not a relative minimum is unstable. For formulas of one-step numerical integration, an estimate of the step of integration is given, so that the above mentioned qualitative properties of the system of ordinary differential equations are kept.     

A cooperative strategy for solving dynamic optimization problems

Optimization in dynamic environments is a very active and important area which tackles problems that change with time (as most real-world problems do). In this paper we present a new centralized cooperative strategy based on trajectory methods (tabu search) for solving Dynamic Optimization Problems (DOPs). Two additional methods are included for comparison purposes. The first method is a Particle Swarm Optimization variant with multiple swarms and different types of particles where there exists an implicit cooperation within each swarm and competition among different swarms. The second method is an explicit decentralized cooperation scheme where multiple agents cooperate to improve a grid of solutions. The main goals are: firstly, to assess the possibilities of trajectory methods in the context of DOPs, where populational methods have traditionally been the recommended option; and secondly, to draw attention on explicitly including cooperation schemes in methods for DOPs. The results show how the proposed strategy can consistently outperform the results of the two other methods.     

A new approach for solving repair limit problems

The usual approach to finding optimal repair limits on failure of a component is to use a finite state approximation Markov Decision Process (MDP). In this paper an alternative approach is introduced. Assuming a stochastically increasing repair cost, the optimum solution is shown to satisfy a certain two-point boundary condition, first order differential equation. An asymptotic formula for the optimal repair limit function is derived. Numerical solutions are obtained for some Weibull and Special Erlang distributed time to failure distributions. The structural form of the repair limit function results in a solution procedure which is several orders of magnitude faster than is achievable using previous methods.