Optimization of the two-dimensional temperature fields in inhomogeneous bodies with constraints on the phase coordinates

We state a problem of speed-optimal control of the two-dimensional nonstationary temperature regime in inhomogeneous bodies with constraints on the control (the temperature of the heating medium) and the phase coordinates (the temperature of the body, the temperature range, and the heat flux on the surface of the body), and we propose a method for solving the problem numerically. We give as an example the computation of the optimal control of heating of a hollow infinite cylinder under constraints on the temperature range. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 119–123.     

Optimization of the heating of a heat-sensitive layer under constraints on the stresses in the plastic region of deformation of the material

We study the problem of optimal control for rapidity of the heating of a heat-sensitive layer under constraints on the control (the temperature of the heating medium or the heat flux) and maximal values of the stress intensity in the plastic region of deformation of the material. We propose an algorithm for solving the problem that presumes it has been reduced to the inverse problem of thermoplasticity. For the case of one-sided heating we give a numerical analysis of the direct and inverse problems of thermoplasticity. Translated fromMatematichni Metody i Fiziko-Mekhanichni Polya, Vol. 38, 1995.     

A nonlinear optimal control problem for the nonsteady temperature of a layer with constraints on the thermal stress

We propose a numerical method of constructing the optimal heating regime for a thermally stressed unbounded layer with constraints on the control and thermal stresses. Solving the nonlinear optimization problem for rapidity is reduced to solving the inverse problem of thermoelasticity. The results of numerical studies are presented. Translated fromMatematichni Metody i Fiziko-Mekhanichni Polya, Vol. 38, 1995.     

Speed-optimal control of heating of inhomogeneous bodies by internal heat sources with constraints on the parameters of the thermal process

We use the method of the inverse heat-conduction problem to state and construct a numerical solution of the problem of speed-optimal control of the one-dimensional nonstationary thermal regimes of inhomogeneous bodies using internal heat sources with constraints on the control and the parameters of the thermal process. The control problem is nonlinear since the maximal values of the constraining parameters are attained at internal points of the body whose position changes during heating. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 110–114.     

Simple exact solutions applicable to microwave heating

Summary There is presently considerable interest in the utilization of microwave heating to novel industrial applications. Mathematically such problems involve Maxwell's equations coupled with the heat equation and for which all thermal, electrical and magnetic properties of the material are nonlinearly dependent upon temperature. Accordingly such problems are highly complex and very little theoretical work has been undertaken. The purpose of this paper is to obtain simple exact solutions applicable to microwave heating in the simplest situation, involving only one spatial dimension and assuming that all thermal, electrical and magnetic properties exhibit a power law dependence on temperature. Similarity solutions and other special solutions are examined. These generally result in highly nonlinear coupled systems of ordinary differential equations and although some new closed results are obtained in special cases, in general, such complex systems of ordinary differential equations need to be solved numerically. Roughly speaking, we show that stretching similarity solutions exist only if the power law electrical and thermal conductivities and magnetic permeability with indicesl, m andn respectively are such thatl+m+n=0. Similar constraints on the indices apply for the existence of other simple solutions.     

Local solvability of control systems with mixed constraints

We analyze the local solvability of a control system with mixed constraints and control constraints. We obtain a sufficient condition for the local solvability of the control system under the assumption of smoothness and 2-regularity of the mapping g specifying the mixed constraints. For the case in which the mapping g is not smooth, a sufficient condition for the local solvability is obtained under the assumption of coverability of g.     

On global warming: Flow-based soft global constraints

In case a CSP is over-constrained, it is natural to allow some constraints, called soft constraints, to be violated. We propose a generic method to soften global constraints that can be represented by a flow in a graph. Such constraints are softened by adding violation arcs to the graph and then computing a minimum-weight flow in the extended graph to measure the violation. We present efficient propagation algorithms, based on different violation measures, achieving domain consistency for the alldifferent constraint, the global cardinality constraint, the regular constraint and the same constraint. This work was for a large part carried out while the author was at the Centrum voor Wiskunde en Informatica, Amsterdam, The Netherlands     

Rational regimes for induction heating of cylindrical articles for graphitation

We develop a method of computing regimes for induction heating of cylindrical carbon devices during graphitation. We study the electromagnetic, temperature, and mechanical fields as functions of the parameters of the heating regime.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 33, 1991, pp. 27–31.     

Projection of diffusions on submanifolds: Application to mean force computation

We consider the problem of sampling a Boltzmann‐Gibbs probability distribution when this distribution is restricted (in some suitable sense) on a submanifold Σ of ?ⁿ implicitly defined by N constraints q₁(x) = ? = q_N(x) = 0 (N < n). This problem arises, for example, in systems subject to hard constraints or in the context of free energy calculations. We prove that the constrained stochastic differential equations (i.e., diffusions) proposed in [7, 13] are ergodic with respect to this restricted distribution. We also construct numerical schemes for the integration of the constrained diffusions. Finally, we show how these schemes can be used to compute the gradient of the free energy associated with the constraints. © 2007 Wiley Periodicals, Inc.     

Multiple crane scheduling in a batch annealing process with no-delay constraints for machine unloading

In this work, we focus on the scheduling of multi-crane operations in an iron and steel enterprise for a two-stage batch annealing process. The first stage is the heating process, and the second stage is the cooling process. To start the heating (cooling) stage, a special machine called a furnace (cooler) must be loaded. The real constraints of no-delay machine unloading are defined as follows: once the heating (cooling) is completed, the furnace (cooler) must be unloaded by crane immediately. The goal is to schedule limited machines (furnaces and coolers) operated by multiple cranes to minimize the completion time of the last annealed coil (makespan). We formulate a mixed-integer linear programming model to address this problem. Certain feasible properties are identified to avoid crane conflicts and ensure that the machine unloading no-delay constraints are met. Based on these necessary conditions, we then present a heuristic algorithm with running time in connection with the number of cranes, coils and machines. A lower bound to the problem is also developed. Through theoretical analysis, we show the worst-case bound of our heuristic algorithm. The average performances of the solution approaches are computationally evaluated. The computational results show that the proposed heuristic algorithm is capable of generating good quality solutions.     

Convergence results for the indifference value based on the stability of BSDEs

We study the exponential utility indifference value h for a contingent claim H in an incomplete market driven by two Brownian motions. The claim H depends on a non-tradable asset variably correlated with the traded asset available for hedging. We provide an explicit sequence that converges to h, complementing the structural results for h known from the literature. Our study is based on a convergence result for quadratic backward stochastic differential equations. This convergence result, which we prove in a general continuous filtration under weak conditions, also yields that the indifference value in a setting with trading constraints enjoys a continuity property in the constraints.     

The Economic Lot Scheduling Problem without Capacity Constraints

This study presents a comprehensive analysis on the Economic Lot Scheduling Problem (ELSP) without capacity constraints. We explore the optimality structure of the ELSP without capacity constraints and discover that the curve for the optimal objective values is piecewise convex with repsect to B, i.e., the values of basic period. The theoretical properties of the junction points on the piecewise convex curve not only provides us the information on “which product i” to modify, but also on “where on the B-axis” to change the set of optimal multpliers in the search process. By making use of the junction points, we propose an effective search algorithm to secure a global optimal solution for the ELSP without capacity constraints. Also, we use random experiments to verify that the proposed algorithm is efficient. The results in this paper lay important foundation for deriving an efficient heuristic to solve the conventional ELSP with capacity constraints.     

α-Well-posedness for Nash Equilibria and For Optimization Problems with Nash Equilibrium Constraints

We present the concepts of α-well-posedness for parametric noncooperative games and for optimization problems with constraints defined by parametric Nash equilibria. We investigate some classes of functions that ensure these types of well-posedness and the connections with α-well-posedness for variational inequalities and optimization problems with variational inequality constraints.     

The Cauchy Problem on a Characteristic Cone for the Einstein Equations in Arbitrary Dimensions

We derive explicit formulae for a set of constraints for the Einstein equations on a null hypersurface, in arbitrary space–time dimensions n + 1 ≥ 3. We solve these constraints and show that they provide necessary and sufficient conditions so that a spacetime solution of the Cauchy problem on a characteristic cone for the hyperbolic system of the reduced Einstein equations in wave-map gauge also satisfies the full Einstein equations. We prove a geometric uniqueness theorem for this Cauchy problem in the vacuum case.     

Singular manifolds in optimal control problems

We propose a method of solving an optimal control problem with constraints on the control. The method is applied to find an upper bound of the Hamilton-Pontryagin function and is based on the construction of a system of differential equations containing the constraints available in the problem as singular manifolds. Bibliography: 2 titles. Translated fromProblemy Matematicheskoi Fiziki, 1998, pp. 214–216.     

Constraint classification in mathematical programming

Consider a set of algebraic inequality constraints defining either an empty or a nonempty feasible region. It is known that each constraint can be classified as either absolutely strongly redundant, relatively strongly redundant, absolutely weakly redundant, relatively weakly redundant, or necessary. We show that is is worth making a distinction between weakly necessary constraints and strongly necessary constraints. We also present afeasible set cover method which can detect both weakly and strongly necessary constraints.The main interest in constraint classification is due to the advantages gained by the removal of redundant constraints. Since classification errors are likely to occur, we examine how the removal of a single constraint can affect the classification of the remaining constraints.     

Interval propagation and search on directed acyclic graphs for numerical constraint solving

The fundamentals of interval analysis on directed acyclic graphs (DAGs) for global optimization and constraint propagation have recently been proposed in Schichl and Neumaier (J. Global Optim. 33, 541–562, 2005). For representing numerical problems, the authors use DAGs whose nodes are subexpressions and whose directed edges are computational flows. Compared to tree-based representations [Benhamou et al. Proceedings of the International Conference on Logic Programming (ICLP’99), pp. 230–244. Las Cruces, USA (1999)], DAGs offer the essential advantage of more accurately handling the influence of subexpressions shared by several constraints on the overall system during propagation. In this paper we show how interval constraint propagation and search on DAGs can be made practical and efficient by: (1) flexibly choosing the nodes on which propagations must be performed, and (2) working with partial subgraphs of the initial DAG rather than with the entire graph. We propose a new interval constraint propagation technique which exploits the influence of subexpressions on all the constraints together rather than on individual constraints. We then show how the new propagation technique can be integrated into branch-and-prune search to solve numerical constraint satisfaction problems. This algorithm is able to outperform its obvious contenders, as shown by the experiments.     

Crossing-constrained hierarchical drawings

We study the problem of computing hierarchical drawings of layered graphs when some pairs of edges are not allowed to cross. We show that deciding the existence of a drawing satisfying at least k non-crossing constraints from a given set is NP-hard, even if the graph is 2-layered and even when the permutation of the vertices on one side of the bipartition is fixed. We then propose simple constant-ratio approximation algorithms for the optimization version of the problem, which requires to find a maximum realizable subset of constraints, and we discuss how to extend the well-known hierarchical approach for creating layered drawings of directed graphs so as to minimize the number of edge crossings while maximizing the number of satisfied constraints.     

On Variational Based Scale-Bridging of Inelastic Composites

The paper discusses formulations for the theoretical and numerical analysis of inelastic composites with scale separation. The basic underlying structure is a canonical variational setting in the fully nonlinear range based on incremental energy minimization. We focus on formulations of strain–driven homogenization for representative composite aggregates with emphasis on the development of canonical families of algorithms based on Lagrange and penalty functionals to cover alternative boundary constraints of (i.) linear deformations, (ii.) periodic deformations and (iii.) uniform tractions. As a key result, we present a compact matrix formulation for homogenization covering introduced alternative boundary constraints. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The elastoplastic stressed state of a half-plane under local nonstationary heating

We solve the thermoplastic problem for a semi-infinite plate under local nonstationary heating by heat sources. The physical equations are taken to be the relations of the nonisothermic theory of plastic flow associated with the Mises fluidity condition. The solution of the problem is constructed by the method of integral equations and the self-correcting method of sequential loading, where time is taken as the loading parameter. We carry out numerical computations of the stresses in the case of heating a plate with heat output by normal-circular heat sources. We study the problem of optimization of heating regimes in order to introduce favorable residual compressive stresses (from the point of view of hardness) in a given region of a half-plane. Two figures.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 29–34.