Showing papers in "Journal of Optimization Theory and Applications in 1985"

Thermodynamical approach to the traveling salesman problem: An efficient simulation algorithm

Abstract: We present a Monte Carlo algorithm to find approximate solutions of the traveling salesman problem. The algorithm generates randomly the permutations of the stations of the traveling salesman trip, with probability depending on the length of the corresponding route. Reasoning by analogy with statistical thermodynamics, we use the probability given by the Boltzmann-Gibbs distribution. Surprisingly enough, using this simple algorithm, one can get very close to the optimal solution of the problem or even find the true optimum. We demonstrate this on several examples. We conjecture that the analogy with thermodynamics can offer a new insight into optimization problems and can suggest efficient algorithms for solving them.

Necessary and sufficient conditions for quadratic stabilizability of an uncertain system

Abstract: Consider an uncertain system (Σ) described by the equationx(t)=A(r(t))x(t)+B(s(t))u(t), wherex(t) ∈R n is the state,u(t) ∈R m is the control,r(t) ∈ ℛ ⊂R p represents the model parameter uncertainty, ands(t) ∈L ⊂R l represents the input connection parameter uncertainty. The matrix functionsA(·),B(·) are assumed to be continuous and the restraint sets ℛ,L are assumed to be compact. Within this framework, a notion of quadratic stabilizability is defined. It is important to note that this type of stabilization is robust in the following sense: The Lyapunov function and the control are constructed using only the bounds ℛ,L. Much of the previous literature has concentrated on a fundamental question: Under what conditions onA(·),B(·), ℛ,L can quadratic stabilizability be assured? In dealing with this question, previous authors have shown that, if (Σ) satisfies certain matching conditions, then quadratic stabilizability is indeed assured (e.g., Refs. 1–2). Given the fact that matching is only a sufficient condition for quadratic stabilizability, the objective here is to characterize the class of systems for which quadratic stabilizability can be guaranteed.

The Essence of Invexity

Abstract: The notion of invexity was introduced into optimization theory by Hanson in 1981 as a very broad generalization of convexity. A smooth mathematical program of the form minimizef(x), subject tog(x) ≦ 0, isx ∈D ⊑ ℝ n invex if there exists a function η:D ×D → ℝ n such that, for allx, u ∈D, $$\begin{gathered} f(x) - f(u) - f'(u)n(x,u) \geqq 0, \hfill \\ g(x) - g(u) - g'(u)n(x,u) \geqq 0. \hfill \\ \end{gathered}$$ The convex case corresponds of course to η(x, u)≡x−u; but, as Hanson showed, invexity is sufficient to imply both weak duality and that the Kuhn-Tucker conditions are sufficient for global optimality. It is shown here that elementary relaxations of the conditions defining invexity lead to modified invexity notions which are both necessary and sufficient for weak duality and Kuhn-Tucker sufficiency.

Global optimization and stochastic differential equations

Abstract: Let ℝ n be then-dimensional real Euclidean space,x=(x 1,x 2, ,x n)T ∈ ℝ n , and letf:ℝ n → R be a real-valued function We consider the problem of finding the global minimizers off A new method to compute numerically the global minimizers by following the paths of a system of stochastic differential equations is proposed This method is motivated by quantum mechanics Some numerical experience on a set of test problems is presented The method compares favorably with other existing methods for global optimization

An Algorithm for Generalized Fractional Programs

Abstract: An algorithm is suggested that finds the constrained minimum of the maximum of finitely many ratios. The method involves a sequence of linear (convex) subproblems if the ratios are linear (convex-concave). Convergence results as well as rate of convergence results are derived. Special consideration is given to the case of (a) compact feasible regions and (b) linear ratios.

An example of numerical nonconvergence of a variable-metric method

Abstract: An example is given where truncation error, caused by finite computer arithmetic, leads to the BFGS variable-metric method becoming stuck, despite the approximated Hessian matrix, the gradient vector, and the search vector satisfying analytical conditions for convergence. A restart criterion to eliminate the problem is suggested.

Tractable classes of nonzero-sum open-loop Nash differential games: Theory and examples

Abstract: This paper identifies some classes ofN-person nonzero-sum differential games that are tractable, in the sense that open-loop Nash strategies can be determined, either explicitly or qualitatively in terms of a phase-diagram portrait. The classes are characterized by conditions imposed on the Hamiltonians. Also, the underlying game structures needed to satisfy these conditions are characterized.

A discrete method of optimal control based upon the cell state space concept

Abstract: A discrete method of optimal control is proposed in this paper. The continuum state space of a system is discretized into a cell state space, and the cost function is discretized in a similar manner. Assuming intervalwise constant controls and using a finite set of admissible control levels (u) and a finite set of admissible time intervals (τ), the motion of the system under all possible interval controls (u, τ) can then be expressed in terms of a family of cell-to-cell mappings. The proposed method extracts the optimal control results from these mappings by a systematic search, culminating in the construction of a discrete optimal control table.

Eigenweight vectors and least-distance approximation for revealed preference in pairwise weight ratios

Abstract: A new eigenweight vector is derived for the data of pairwise weight ratios. The well-known eigenweight vector derived by Saaty is then compared and contrasted in the light of least-distance approximation models. It is shown that the new eigenweight vector commands advantages over Saaty's, including less rigid assumptions on the error terms, robustness of solution, in addition to the fact that the new eigenweight vector can be computed very easily. The reader can construct other types of eigenweight vectors and least-distance approximation models using the framework of this article.

Shortest paths in networks with vector weights

Abstract: For a directed network in which vector weights are assigned to arcs, the Pareto analog to the shortest path problem is analyzed. An algorithm is presented for obtaining all Pareto shortest paths from a specified node to every other node.

Dynamic common property resources and environmental problems

Abstract: Problems of interacting common-property resources are set up as stochastic differential games. A class of models is solved where equilibrium closed-loop strategies keep harvest rates proportional to stocks. Corrective taxes, etc., are considered.

Directional derivatives in nonsmooth optimization

Abstract: In this note, we consider two notions of second-order directional derivatives and discuss their use in the characterization of minimal points for nonsmooth functions.

Lower subdifferentiable functions and their minimization by cutting planes

Abstract: This paper introduces lower subgradients as a generalization of subgradients. The properties and characterization of boundedly lower subdifferentiable functions are explored. A cutting plane algorithm is introduced for the minimization of a boundedly lower subdifferentiable function subject to linear constraints. Its convergence is proven and the relation is discussed with the well-known Kelley method for convex programming problems. As an example of application, the minimization of the maximum of a finite number of concave-convex composite functions is outlined.

Optimal control of delay systems via block pulse functions

Abstract: The concept of coefficient shift matrix is introduced to represent delay variables in block pulse series. The optimal control of a linear delay system with quadratic performance index is then studied via block pulse functions, which convert the problems into the minimization of a quadratic form with linear algebraic equation constraints. The solution of the two-point boundary-value problem with both delay and advanced arguments is circumvented. The control variable obtained is piecewise constant.

Games with vector payoffs

Abstract: Two-person games are defined in which the payoffs are vectors. Necessary and sufficient conditions for optimal mixed strategies are developed, and examples are presented.

Existence of equilibrium stationary strategies in discounted noncooperative stochastic games with uncountable state space

Abstract: This paper considers discounted noncooperative stochastic games with uncountable state space and compact metric action spaces. We assume that the transition law is absolutely continuous with respect to some probability measure defined on the state space. We prove, under certain additional continuity and integrability conditions, that such games have e-equilibrium stationary strategies for each e>0. To prove this fact, we provide a method for approximating the original game by a sequence of finite or countable state games. The main result of this paper answers partially a question raised by Parthasarathy in Ref. 1.

Optimal control computation for nonlinear time-lag systems

Abstract: In this paper, a computational scheme using the technique of control parameterization is developed for solving a class of optimal control problems involving nonlinear hereditary systems with linear control constraints. Several examples have been solved to test the efficiency of the technique.

On stochastic games with additive reward and transition structure

Abstract: In this paper, we introduce a new class of two-person stochastic games with nice properties. For games in this class, the payoffs as well as the transitions in each state consist of a part which depends only on the action of the first player and a part dependent only on the action of the second player. For the zero-sum games in this class, we prove that the orderfield property holds in the infinite-horizon case and that there exist optimal pure stationary strategies for the discounted as well as the undiscounted payoff criterion. For both criteria also, finite algorithms are given to solve the game. An example shows that, for nonzero sum games in this class, there are not necessarily pure stationary equilibria. But, if such a game possesses a stationary equilibrium point, then there also exists a stationary equilibrium point which uses in each state at most two pure actions for each player.

Impulsive optimal control with finite or infinite time horizon

Abstract: We consider a dynamical system subjected to feedback optimal control in such a way that the evolution of the state exhibits both sudden jumps and continuous changes. Previously obtained necessary conditions (Ref. 1) for such impulsive optimal feedback controls are generalized to admit the case of infinite time horizon; this generalization permits application to a wider class of problems. The results are illustrated by application to a version of the innkeeper's problem.

Optimization of properties of multicomponent isotropic composites

Abstract: We suggest a method for the calculation of the extremal conductivity of composites under some natural assumptions concerning their microstructure. The method is based on the principle of consecutive assembling of binary mixtures by addition of infinitely small amounts of one of the initial compounds to the already-assembled isotropic composite. This process is assumed to produce an optimal isotropic binary mixture at each step, which is performed by the Hashin-Shtrikman procedure. We are seeking a suitable sequence of compounds to be added to the mixture in order to minimize its resultant conductivity. A solution is given to the corresponding optimization problems for both finite number and infinite number of initial compounds taken in prescribed concentrations. We also describe the microstructure of the optimal composites. The results can be used for the optimal design of elastic and heat-conducting constructions.

Practical Stabilizability of Uncertain Dynamical Systems: Application to Robotic Tracking ~

Abstract: The tracking problem for a robotic manipulator withn controlled degrees of freedom and uncertain dynamics is considered. Based on the deterministic theory of Refs. 1–15 and requiring only knowledge of bounds on the system uncertainty, a classC of continuous feedback controls (adapted from Ref. 11) is proposed with respect to which the uncertain tracking system is practically stabilizable in the sense that, given a feasible path to be tracked and an arbitrarily small neighborhood σ of the origin in the appropriate error space, there exists a control inC such that the tracking error for the feedback controlled uncertain system is ultimately bounded with respect to σ. The theory is illustrated in a numerical example of a robot with two degrees of freedom.

On the existence of time-optimal control of mechanical manipulators

Abstract: This paper is concerned with the problem of the existence and structure of time-optimal control for models derived from Lagrange equations of motion of mechanical systems involving links. The condition which ensures the existence of time-optimal control is demonstrated. The study conducted in this paper involves a highly nonlinear mathematical model of a two-degree-of-freedom mechanical system. However, the procedure and the results presented in this paper can be extended to mechanical systems with any finite number of degrees of freedom.

On differential games of evasion from many pursuers

Abstract: This paper contains a survey of some results regarding differential games of evasion from many pursuers. This class of games presents special difficulties and usually cannot be treated by standard methods. The approach developed consists of constructing piecewise program strategies for the evader, based on certain maneuvers of evasion from one pursuer. These strategies satisfy one additional condition (state constraint): the evader's motion does not leave a given neighborhood of a prescribed nominal motion. An upper estimate for the number of program pieces of the evader's control and a lower estimate for the minimal distance between the evader and the pursuers are also obtained. These results are given for several types of equations of the game.

Adaptive control of discounted Markov decision chains

Abstract: In this paper, we consider discounted-reward finite-state Markov decision processes which depend on unknown parameters. An adaptive policy inspired by the nonstationary value iteration scheme of Federgruen and Schweitzer (Ref. 1) is proposed. This policy is briefly compared with the principle of estimation and control recently obtained by Schal (Ref. 4).

On the structure of the set bases of a degenerate point

Abstract: Consider an extreme point (EP)x 0 of a convex polyhedron defined by a set of linear inequalities. If the basic solution corresponding tox 0 is degenerate,x 0 is called a degenerate EP. Corresponding tox 0, there are several bases. We will characterize the set of all bases associated withx 0, denoted byB 0. The setB 0 can be divided into two classes, (i) boundary bases and (ii) interior bases. For eachB 0, there is a corresponding undirected graphG 0, in which there exists a tree which connects all the boundary bases. Some other properties are investigated, and open questions for further research are listed, such as the connection between the structure ofG 0 and cycling (e.g., in linear programs).

A new three-term conjugate gradient method

Abstract: In this paper, we describe an implementation and give performance results for a conjugate gradient algorithm for unconstrained optimization. The algorithm is based upon the Nazareth three-term formula and incorporates Allwright preconditioning matrices and restart tests. The performance results for this combination compare favorably with existing codes.

Efficient spanning trees

Abstract: The definition of a shortest spanning tree of a graph is generalized to that of an efficient spanning tree for graphs with vector weights, where the notion of optimality is of the Pareto type. An algorighm for obtaining all efficient spanning trees is presented.

Definition and properties of cooperative equilibria in a two-player game of infinite duration

Abstract: A two-player multistage game, with an infinite number of stages is considered. The concepts of overtaking and weakly overtaking payoff sequences are introduced. The class of strategies considered consists of memory strategies, which are based on the past history of the control and the initial state from where the game has been played. Weak equilibria are defined in this class of strategies. It is then shown how such equilibria can be constructed by composing into a trigger strategy a nominal cooperative control sequence and two threat strategies representing the announced retaliation by each player in the case where the other player does not play according to the nominal control. When the threats consists of a feedback equilibrium pair, the resulting cooperative equilibrium is perfect. Another result shows that, if each player can use a most effective threat based on a saddle-point feedback strategy, then any weak equilibrium in the class of memory strategies is in some sense related to this particular kind of equilibrium in the class of trigger strategies.

Topological methods for the global controllability of nonlinear systems

Abstract: Sufficient conditions for the local and global controllability of general nonlinear systems, by means of controls belonging to a fixed finite-dimensional subspace of the space of all admissible controls, are established with the aid of topological methods, such as homotopy invariance principles. Some applications to certain classes of nonlinear control processes are given, and various known results on the controllability of perturbed linear systems are also derived as particular cases.

Optimal quasi-static shape control for large aerospace antennae

Abstract: In this paper, we propose an on-line control approach which will adjust the steady-state shape of a large antenna arbitrarily close to any achievable desired profile. The method makes use of distributed parameter system theory and allows refocusing using a limited number of control actuators and sensors. The controller gains are calculated by approximating the solution to an infinite-dimensional optimal quasi-static control problem. We prove a very general convergence result for such quasi-static controllers here and apply it to the antenna controller to show convergence using any Galerkin (finite-element) approximation method.