An optimal design problem with perimeter penalization

We study the optimal design problem of finding the minimal energy configuration for a mixture of two conducting materials when a perimeter penalization of the unknown domain is added. We show that in this situation an optimal domain exists and that, under suitable assumptions on the data, it is an open set.This work is part of the project EURHomogenization, contract SC1-CT91-0732 of the program SCIENCE of the Commission of the European Communities.     

An optimal capacity assignment for the robust design problem in capacitated flow networks

This paper proposes an exact algorithm to solve the robust design problem in a capacitated flow network in which each edge has several possible capacities. A capacitated flow network is popular in our daily life. For example, the computer network, the power transmission network, or even the supply chain network are capacitated flow networks. In practice, such network may suffer failure, partial failure or maintenance. Therefore, each edge in the network should be assigned sufficient capacity to keep the network functioning normally. The robust design problem (RDP) in a capacitated flow network is to search for the minimum capacity assignment of each edge such that the network still survived even under the edge’s failure. However, how to optimally assign the capacity to each edge is not an easy task. Although this kind of problem was known of NP-hard, this paper proposes an efficient exact algorithm to search for the optimal solutions for such a network and illustrates the efficiency of the proposed algorithm by numerical examples.     

An optimal stopping problem for fragmentation processes

In this article we consider a toy example of an optimal stopping problem driven by fragmentation processes. We show that one can work with the concept of stopping lines to formulate the notion of an optimal stopping problem and moreover, to reduce it to a classical optimal stopping problem for a generalized Ornstein–Uhlenbeck process associated with Bertoin’s tagged fragment. We go on to solve the latter using a classical verification technique thanks to the application of aspects of the modern theory of integrated exponential Lévy processes.     

An optimal control problem for microwave heating

In this paper, we study an optimization problem for a microwave/induction heating process. The cost function is defined such that the temperature profile at the final stage has a relative uniform distribution in the field. The control variable is the applied electric field on the boundary. We show that there exists an optimal electric field which minimizes the cost function. Moreover, a necessary condition for a special case is also derived.     

An implementable algorithm for the optimal design centering,tolerancing, and tuning problem

An implementable master algorithm for solving optimal design centering, tolerancing, and tuning problems is presented. This master algorithm decomposes the original nondifferentiable optimization problem into a sequence of ordinary nonlinear programming problems. The master algorithm generates sequences with accumulation points that are feasible and satisfy a new optimality condition, which is shown to be stronger than the one previously used for these problems.This research was sponsored by the National Science Foundation (RANN), Grant No. ENV-76-04264, and by the Joint Services Electronic Program, Contract No. F44620-76-C-0100.     

An optimal control problem for the Hoff equation

The sufficient and necessary conditions are given for existence of an optimal control in the bending problem for an I-beam.     

二阶数乘问题的一个最优算法

研究一个有趣的组合优化问题——二阶数乘问题.问题描述如下：给定n≥2个正整数a_1,a_2,…,a_n,设π为{1,2,…,n}的一个置换,表示该问题的一个解,试图找到一个置换π以至∑_（i=1）~n a_（π_i）a_（π_（i＋1））最小,在这里π_（n＋1）=π_1.给出了一个算法复杂度为O（n log n）的最优算法.     

An optimal lower bound for the Frobenius problem

Given N?2 positive integers a₁,a₂,…,aN with GCD(a₁,…,aN)=1, let fN denote the largest natural number which is not a positive integer combination of a₁,…,aN. This paper gives an optimal lower bound for fN in terms of the absolute inhomogeneous minimum of the standard (N−1)-simplex.     

On the unique solvability of a problem on water waves above submerged bodies

Time-harmonic motion of an ideal unbounded fluid in the presence of rigid bodies located under fluid’s free surface is considered. New criteria for the unique solvability of the corresponding linear boundary-value problem are suggested. These criteria are based on the introduction of two compact self-adjoint integral operators and on the investigation of their eigenvalues and eigenfunctions. For the two-dimensional problem, an algorithm is developed for finding solutions to the homogeneous problem (the so-called trapped modes). Examples of numerical computations illustrating the theoretical results are given. Bibliography: 18 titles.     

An optimal problem in graph theory

The following problem is solved: determine a point on a tree having the property that the sum of the products of the intensities of its vertices by the corresponding distances to that point is a minimum. The proposed algorithm is reduced to the stepwise application to the tree of truncation of its vertices. A feasible interpretation of the problem is given.Translated from Matematicheskie Zametki, Vol. 10, No. 3, pp. 355–359, September, 1971.     

An optimal algorithm for the minimum-width cubic shell problem

Given a set of n points in ${\mathbb{R}}^3$, the minimum-width cubic shell problem asks to find a thinnest cubic shell that encloses the input points, where a cubic shell refers to as a closed volume between two concentric axis-aligned cubes. In this paper, we improve the previous $O(n{\log }^{2}n)$-time algorithm presented in Bae (2019) [6] to $O(n\log n)$ worst-case time. This is the first optimal-time algorithm to the problem.     

An iterative scheme for solving the optimal transportation problem

We demonstrate an iterative scheme to approximate the optimal transportation problem with a discrete target measure under certain standard conditions on the cost function. Additionally, we give a finite upper bound on the number of iterations necessary for the scheme to terminate, in terms of the error tolerance and number of points in the support of the discrete target measure.     

An optimal packing problem for satellites in geosynchronous orbit

This paper describes a physical system for which an optimization method is needed. First, the physical nature of the problem is described, and then a mathematical model is developed. The purpose of this paper is to encourage mathematical and control systems analysts to tackle this problem.     

On an optimal control problem for the shape of thin inclusions in elastic bodies

An optimal control problem is considered for a two-dimensional elastic body with a straight thin rigid inclusion and a crack adjacent to it. It is assumed that the thin rigid inclusion delaminates and has a kink. On the crack faces the boundary conditions are specified in the form of equalities and inequalities which describe the mutual nonpenetration of the crack faces. The derivative of the energy functional along the crack length is used as the objective functional, and the position of the kink point, as the control function. The existence is proved of the solution to the optimal control problem.     

Regularity results for an optimal design problem with quasiconvex bulk energies

Regularity results for equilibrium configurations of variational problems involving both bulk and surface energies are established. The bulk energy densities are uniformly strictly quasiconvex functions with quadratic growth, but are otherwise not subjected to any further structure conditions. For a minimal configuration (u, E), partial Hölder continuity of the gradient of the deformation u is proved, and partial regularity of the boundary of the minimal set E is obtained.     

A convergent adaptive finite element method for an optimal design problem

The optimal design problem for maximal torsion stiffness of an infinite bar of given geometry and unknown distribution of two materials of prescribed amounts is one model example in topology optimisation. It eventually leads to a degenerate convex minimisation problem. The numerical analysis is therefore delicate for possibly multiple primal variables u but unique derivatives σ : = DW(D u). Even fine a posteriori error estimates still suffer from the reliability-efficiency gap. However, it motivates a simple edge-based adaptive mesh-refining algorithm (AFEM) that is not a priori guaranteed to refine everywhere. Its convergence proof is therefore based on energy estimates and some refined convexity control. Numerical experiments illustrate even nearly optimal convergence rates of the proposed AFEM. Supported by the DFG Research Center MATHEON “Mathematics for key technologies” in Berlin.     

An optimal control problem in cancer chemotherapy

We consider a mathematical model for the control of the growth of tumor cells which is formulated as a problem of optimal control theory. It is concerned with chemotherapeutic treatment of cancer and aims at the minimization of the size of the tumor at the end of a certain time interval of treatment with a limited amount of drugs. The treatment is controlled by the dosis of drugs that is administered per time unit for which also a limit is prescribed. It is shown that optimal controls are of bang-bang type and can be chosen at the upper limit, if the total amount of drugs is large enough.