On the existence of solutions to a class of optimization problems

A sufficient condition for the existence and uniqueness of solutions to a class of optimization problems in nonlinear programming form, with strictly convex cost functions, convex inequality and linear equality side constraints, and closed convex constraint sets is studied.     

On a class of pursuit-evasion problems

This paper shows that a pursuit-evasion problem can be made amenable to solution with nonlinear programming algorithms by operating the pursuer and evader systems in a "discrete" mode of control and by choosing the cost function judiciously.     

On a class of optimization problems for framed structures

A unified treatment is presented for four types of problems on limit analysis of framed structures and related design problems. With the use of the lower bound theorem in plasticity these problems are formulated as standard linear programming problems. Two significant improvements are made. The new formulation of the design problems reduces the number of equations in the resulting linear program. An improved simplex algorithm for large and sparse linear programs is employed. The formulational and algorithmic improvements provide large capacity and high efficiency that are indispensable for computational analysis and design of large and complex structures.     

An efficient numerical algorithm for the solution of a class ofoptimal control problems

A numerical algorithm for the solution of a class of optimal control problems is presented. This class is characterized as being linear-quadratic in the control, but not necessarily in the state. The algorithm is efficient in that it possesses a high rate of convergence (like Newton's method), but without being sensitive to initialization. In this respect, it combines the advantages of both the first- and second-order algorithms within the framework of a single algorithm. In addition, the algorithm requires less computation and storage than Newton's method     

The set of efficient solutions for multiple objective shortest path problems

This paper considers an extension of the usual scalar valued shortest path problem to one in which there are m objective functions and for which we wish to find the efficient set of policies and paths. The paper discusses some of the deficiencies involved in the use of weighting factor methods to determine the efficient sets with respect to the original pure paths and policies. The main result is that if we wish to find which original paths are efficient with respect to the convex extensions of such sets, then they may be obtained exactly by the weighting factor method with minimisation restricted to the original pure policies, and such optimising policies will be jointly efficient in the convex extension, and hence in the original pure set. The paper also discusses linear programming and dynamic programming computational aspects.     

Some efficient algorithms for a class of abstract optimization problems arising in optimal control

Three abstract optimization problems are presented along with doubly iterative algorithms for their numerical solution. These algorithms are generalizations of particular algorithms described by Barr and Gilbert [19], [21] and Fujisawa and Yasuda [22]. The supporting theory is fully developed along with proofs of convergence. Practical aspects of computations are considered and procedures which insure rapid convergence are discussed. Two applications to discrete-time optimal control problems are described.     

On a class of enclosure methods for initial value problems

This paper deals with the computation of interval enclosure for the solutions of initial value problems in systems of ordinary differential equations. We present a particular class of enclosure methods based on the Taylor series method. They are expressed in a single algorithm containing some parameters which may be chosen arbitrarily. However, a skillful choice is necessary in order to keep the widths of the enclosing intervals sufficiently small. We give some theoretical results about the behaviour of these widths.     

Complexities of efficient solutions of rectilinear polygon cover problems

The rectilinear polygon cover problem is one in which a certain class of features of a rectilinear polygon ofn vertices has to be covered with the minimum number of rectangles included in the polygon. In particular, we consider covering the entire interior, the boundary, and the set of corners of the polygon. These problems have important applications in storing images and in the manufacture of integrated circuits. Unfortunately, most of these problems are known to be NP-complete. Hence it is necessary to develop efficient heuristics for these problems or to show that the design of efficient heuristics is impossible. In this paper we show:

On forbidden state problems for a class of controlled Petri nets

This paper treats the forbidden state problem for the class of discrete-event dynamical systems (DEDS's) which can be modeled as controlled state machines (CtlSM's). CtlSM's constitute a special class of controlled Petri nets (CtlPN's) where no synchronization requirements are included in the system model. Synchronization and other safety requirements are modeled via sets of forbidden states. In this paper we study methods guaranteeing these constraints by disabling some of the controllable transitions. We assume that the full state is observable. Control logic synthesis taking into account the graphical representation of CtlSM's and the distributed state representation in terms of markings is proposed and justified. Two examples-the cat-and-mouse game and a model of a metro line-illustrate the theoretical results     

On studying monotonicity in the parameter of optimal solutions for one class of the parametric optimization problems

Consideration was given to minimization of a nonnegative nondecreasing function under the linear constraints comprising a scalar parameter t of the right side of the constraints. Monotonicity in the parameter t of the optimal solutions of the considered problem plays an important part in some applied optimization models such as static optimization of the transportation systems with the linear constraints x ≥ 0, Ax = tb. Some possible interpretations (economic, transport-economic, and investment) of the considered mathematical problem were presented, and the problem history was outlined. Relatively simple, yet hard, sufficient conditions for monotonicity were obtained. Simple examples demonstrating that this property may be violated already in the simplest cases were presented.     

An efficient approach to approximate solutions of eighth-order boundary-value problems

In this paper, the approximate solutions to the eighth-order boundary-value differential equations are solved by using the Adomian decomposition method (ADM). The numerical solutions of the problem are calculated in the form of a series with easily computable components. The numerical illustrations show that this technique is more reliable, efficient and accurate than the traditional schemes.     

Optimal solutions to a class of power management problems in mobile robots

This paper studies an approach to minimize the power consumption of a mobile robot by controlling its traveling speed and the frequency of its on-board processor simultaneously. The problem is formulated as a discrete-time optimal control problem with a random terminal time and probabilistic state constraints. A general solution procedure suitable for arbitrary power functions of the motor and the processor is proposed. Furthermore, for a class of realistic power functions, the optimal solution is derived analytically. Interpretations of the optimal solution in the practical context are also discussed. Simulation results show that the proposed method can save a significant amount of energy compared with some heuristic schemes.     

On stability of solutions for a class of nonlinear difference systems with switching

The stability of the trivial solution for a class of difference systems with switching and sector-type nonlinearities is studied. Different approaches to common Lyapunov function design for the family of subsystems corresponding to the considered switched system are proposed. Sufficient conditions making the trivial solution asymptotically stable for any switching law are determined. In the case when common Lyapunov function design fails, multiple Lyapunov functions are used to obtain the restrictions on the switching law guaranteeing the asymptotic stability of the trivial solution.     

On efficient multigrid software for elliptic problems on rectangular domains

Emphasis has been laid on software development concerning ‘non-standard’ multigrid techniques as introduced by Foerster, Stüben and Trottenberg¹. The advantages of these methods result in very efficient code for the solution of elliptic problems. In comparison with fast direct methods, the multigrid solver MGOO is as good as direct solvers for model equations on rectangular domains, even better in special cases, and more generally applicable (variable coefficients).