Maximizing a concave function over the efficient or weakly-efficient set

An important approach in multiple criteria linear programming is the optimization of some function over the efficient or weakly-efficient set. This is a very difficult nonconvex optimization problem, even for the case that the function to be optimized is linear. In this article we consider the problem of maximizing a concave function over the efficient or weakly-efficient set. We show that this problem can essentially be formulated as a special global optimization problem in the space of the extreme criteria of the underlying multiple criteria linear program. An algorithm of branch and bound type is proposed for solving the resulting problem.     

Anisotropic diffusions with singular advections and absorptions part 1: existence

An anisotropic filtration problem with singular advections and strong absorptions, which leads to a quasi-linear degenerate parabolic equation in divergent form, is studied in this paper. The existence for the Cauchy problem as well as the first boundary-initial-value problem is proved.     

A branch and bound network approach to the canonical constrained entropy problem

In this paper, we present a branch and bound algorithm for solving the constrained entropy mathematical programming problem. Unlike other methods for solving this problem, our method solves more general problems with inequality constraints. The advantage of the proposed technique is that the relaxed problem solved at each node is a singly constrained network problem. The disadvantage is that the relaxed problem has twice as many variables as the original problem. An application to regional planning is given, and an example problem is solved.     

Optimal cargo allocation on board a plane: a sequential linear programming approach

We examine a case study of an airline company whose problem is to plan cargo allocations on board a plane. Given the volume, weight, and structural constraints, the problem of finding the optimal load layout is formulated as a fractional programming problem. An algorithm is suggested to solve the linearized problem as a sequence of linear programming problems whose optimal solutions converge to the optimum (with a predetermined level of tolerance).     

An inverse problem for the quasistatic thermoelastic system on the unit Disk

An inverse problem for a quasistatic, linearized, thermoelastic system on the unit disk is formulated as a minimization problem, by use of function theoretic methods and a potential representation.     

Network flows with age dependent decay rates

This paper considers the problem of maximizing the output flow in a multicommodity network in which flow entering an arc experiences a decay rate which is a function of three factors: the arc, the commodity, and the age of the commodity as it enters the arc. An arc-chain linear programming formulation of the problem is given. The algorithm for solving the problem involves a novel column generation scheme for basis entry embedded in the revised simplex algorithm. An efficient algorithm for generating, at each iteration, such a column is provided and illustrated with a numerical example.     

An Algorithm for Solving 3-Dimensional Assignment Problems with Application to Scheduling a Teaching Practice

A scheduling problem associated with teaching practices at colleges of education is formulated as a 3-dimensional assignment problem. An efficient algorithm for its solution, based on Lagrangean relaxation, is described.     

An efficient algorithm for optimal design of area traffic control with network flows

An equilibrium network design (EQND) is a problem of finding the optimal design parameters while taking into account the route choice of users. This problem can be formulated as an optimization by taking the user equilibrium traffic assignment as a constraint. In this paper, the methods solving the EQND problem with signal settings are investigated via numerical calculations on two example road networks. An efficient algorithm is proposed in which improvement on a locally optimal search by combining the technique of parallel tangents with the gradient projection method is presented. As it shows, the method combines the locally optimal search and globally search heuristic achieved substantially better performance than did those other approaches.     

Theoretical framework of an identification problem for an elliptic variational inequality with bilateral restrictions

An identification problem associated to an elliptic variational inequation subject to a bilateral restriction is considered. The whole of the parameters involved in the inequation as well as the parameters defining the restriction are to be identified. The continuous dependence of the direct problem solution on these parameters is proved. As a consequence the well-posedness of the identification problem follows.     

Application of fast automatic differentiation for solving the inverse coefficient problem for the heat equation

The problem of determining the thermal conductivity coefficient that depends on temperature is studied. The consideration is based on the initial-boundary value problem for the one-dimensional unsteady heat equation. The mean-root-square deviation of the temperature distribution field and the heat flux from the experimental data on the left boundary of the domain is used as the objective functional. An analytical expression for the gradient of the objective functional is obtained. An algorithm for the numerical solution of the problem based on the modern fast automatic differentiation technique is proposed. Examples of solving the problem are discussed.     

Approximate Solution of Optimization Problems for Infinite-Dimensional Singular Systems

An optimal control problem is considered for a system described by a singular equation of parabolic type. The study bases on a special regularization method. We establish existence of a solution to the regularized problem, as well as the corresponding necessary optimality conditions. The results enable us to find an approximate solution to the original problem even in the absence of solvability.     

On generalized matrix approximation problem in the spectral norm

In this paper theoretical results regarding a generalized minimum rank matrix approximation problem in the spectral norm are presented. An alternative solution expression for the generalized matrix approximation problem is obtained. This alternative expression provides a simple characterization of the achievable minimum rank, which is shown to be the same as the optimal objective value of the classical problem considered by Eckart–Young–Schmidt–Mirsky, as long as the generalized problem is feasible. In addition, this paper provides a result on a constrained version of the matrix approximation problem, establishing that the later problem is solvable via singular value decomposition.     

A conductive crack and a remote electrode at the interface between two piezoelectric materials

An interaction of a tunnel conductive crack and a distant strip electrode situated at the interface between two piezoelectric semi-infinite spaces is studied. The bimaterial is subject by an in-plane electrical field parallel to the interface and by an anti-plane mechanical loading. Using the presentations of electromechanical quantities at the interface via sectionally-analytic functions the problem is reduced to a combined Dirichlet-Riemann boundary value problem. Solution of this problem is found in an analytical form excepting some one-dimensional integrals calculations. Closed form expressions for the stress, the electric field and their intensity factors, as well as for the crack faces displacement jump are derived. On the base of these presentations the energy release rate is also found. The obtained solution is compared with simple particular case of a single crack without electrode and the excellent agreement is found out. An auxiliary plane problem for open and closed cracks between two isotropic materials is also considered. The mathematical model of this problem is identical to the above one, therefore, the obtained solution is used for this model. It is compared with finite element solution of a similar problem and good agreement is found out.     

An iterative procedure for determining an unknown spacewise-dependent coefficient in a parabolic equation

An inverse problem for the determination of an unknown spacewise-dependent coefficient in a parabolic equation is considered. The problem is reformulated as a nonclassical parabolic equation along with the initial and boundary conditions. The iterative fixed point projection method is applied to solve the reformulated problem. The comparison analysis of proposed method with a least square method and some numerical examples are presented.     

A time-dependent multiple criteria single-machine scheduling problem

We introduce a nonpreemptive single-machine scheduling model with time-dependent multiple criteria. We formulate the problem as a knapsack problem and propose a dynamic programming (DP)-based algorithm to finding all efficient schedules. An illustrative example is enclosed.     

Interval optimal control problem in a Hilbert space

An optimal control problem for a system involving an interval parameter is considered. The concepts of a universal optimal state and a universal optimal control are introduced. The existence and uniqueness of a universal solution to the interval optimal control problem is proved, and an algorithm for its determination is presented. The interval optimal control problem for a system described by the boundary value problem for a second-order ordinary differential equation is solved as an example.     

Solution of the 2-D eddy current problem via the domain decomposition methods on serial and parallel computers

Domain decomposition algorithms are applied to the solution of a time harmonic two-dimensional eddy current problem. The system of differential equations describing this problem is considered as a singularly perturbed problem. An iterative domain decomposition algorithm suitable for parallelization is described, and convergence of this algorithm is established. The implementation on a shared memory multiprocessor is described, and numerical experiments are presented.     

A multi-product loading problem: a model and solution method

An important operational problem arises during the transportation and delivery of several products, which cannot be mixed, in the same vehicle at regular intervals. The vehicle has compartments to keep the products separately. Therefore, a scheme of allocation of compartments which we call vehicle loading problem to maximize the efficiency of the system while the demands for the products at the destination(s) are satisfied. A mixed binary model is developed for this multi-product loading problem. The solution method is based on simultaneously exploring the primal and dual structures derived from the Lagrangian relaxation. Subset sum problems are obtained as subproblems to the partial Lagrangian. An algorithm is developed and its convergence is proved. The efficiency of the method is demonstrated by running, randomly chosen test problems. An initial solution finding method is also developed.     

Frequency planning as a set partitioning problem

Some aspects of a new computerized method for automatic generation of frequency plans for radio communication systems are presented. The emphasis is on problem formulation where the frequency planning problem is recast as a set partitioning problem. The objective is minimization of total system interference. A solution algorithm that has been found useful in practical applications is presented. An alternative algorithm is also demonstrated.