Showing papers in "Journal of Optimization Theory and Applications in 2003"
TL;DR: In this paper, an iterative algorithm is proposed to generate a sequence (x n ✓ n ✓ ) from an arbitrary initial x 0∈H, which converges in norm to the unique solution of the quadratic minimization problem.
Abstract: Assume that C
1, . . . , C
N
are N closed convex subsets of a real Hilbert space H having a nonempty intersection C. Assume also that each C
i
is the fixed point set of a nonexpansive mapping T
i
of H. We devise an iterative algorithm which generates a sequence (x
n
) from an arbitrary initial x
0∈H. The sequence (xn) is shown to converge in norm to the unique solution of the quadratic minimization problem min
x∈C
(1/2)〈Ax, x〉−〈x, u〉, where A is a bounded linear strongly positive operator on H and u is a given point in H. Quadratic–quadratic minimization problems are also discussed.
617 citations
TL;DR: In this paper, a weak convergence theorem for a pair of a nonexpansive mapping and a strictly pseudocontractive mapping was obtained for the problem of finding a common element of the set of fixed points of a nonsmooth mapping and the solutions of a variational inequality problem for a strongly monotone mapping.
Abstract: In this paper, we introduce an iteration process of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality problem for an inverse strongly-monotone mapping, and then obtain a weak convergence theorem. Using this result, we obtain a weak convergence theorem for a pair of a nonexpansive mapping and a strictly pseudocontractive mapping. Further, we consider the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of zeros of an inverse strongly-monotone mapping.
588 citations
TL;DR: In this article, an iterative algorithm is proposed to generate a sequence (xn) from an arbitrary initial point x0∈H, which converges in norm to the unique solution u* of the variational inequality.
Abstract: Assume that F is a nonlinear operator on a real Hilbert space H which is η-strongly monotone and κ-Lipschitzian on a nonempty closed convex subset C of H. Assume also that C is the intersection of the fixed point sets of a finite number of nonexpansive mappings on H. We devise an iterative algorithm which generates a sequence (xn) from an arbitrary initial point x0∈H. The sequence (xn) is shown to converge in norm to the unique solution u* of the variational inequality
$$\left\langle {F(u*),\user1{v} - u*} \right\rangle \geqslant 0$$
Applications to constrained pseudoinverse are included.
289 citations
TL;DR: A novel algorithm for solving multiparametric linear programming problems that follows a geometric approach based on the direct exploration of the parameter space and has computational advantages, namely the simplicity of its implementation in a recursive form and an efficient handling of primal and dual degeneracy.
Abstract: We propose a novel algorithm for solving multiparametric linear programming problems. Rather than visiting different bases of the associated LP tableau, we follow a geometric approach based on the direct exploration of the parameter space. The resulting algorithm has computational advantages, namely the simplicity of its implementation in a recursive form and an efficient handling of primal and dual degeneracy. Illustrative examples describe the approach throughout the paper. The algorithm is used to solve finite-time constrained optimal control problems for discrete-time linear dynamical systems.
219 citations
TL;DR: In this article, an efficient algorithm, called the time-optimal switching (TOS) algorithm, is proposed for the time optimal switching control of nonlinear systems with a single control input, which is formulated in the arc times space, arc times being the durations of the arcs.
Abstract: An efficient algorithm, called the time-optimal switching (TOS) algorithm, is proposed for the time-optimal switching control of nonlinear systems with a single control input. The problem is formulated in the arc times space, arc times being the durations of the arcs. A feasible switching control, or as a special case bang-bang control, is found using the STC method previously developed by the authors to get from an initial point to a target point with a given number of switchings. Then, by means of constrained optimization techniques, the cost being considered as the summation of the arc times, a minimum-time switching control solution is obtained. Example applications of the TOS algorithm involving second-order and third-order systems are presented. Comparisons are made with a well-known general optimal control software package to demonstrate the efficiency of the algorithm.
139 citations
TL;DR: In this paper, the authors consider a general equilibrium problem defined on a convex set, whose cost bifunction may not be monotone, and show that this problem can be solved by the inexact proximal point method if there exists a solution to the dual problem.
Abstract: We consider a general equilibrium problem defined on a convex set, whose cost bifunction may not be monotone. We show that this problem can be solved by the inexact proximal point method if there exists a solution to the dual problem. An application of this approach to nonlinearly constrained problems is also suggested.
134 citations
TL;DR: In this article, several kinds of invariant monotone maps and generalized invariant maps are introduced, and relations between generalized monotonicity and generalized invexity are established, which are generalizations of those presented by Karamardian and Schaible.
Abstract: In this paper, several kinds of invariant monotone maps and generalized invariant monotone maps are introduced. Some examples are given which show that invariant monotonicity and generalized invariant monotonicity are proper generalizations of monotonicity and generalized monotonicity. Relationships between generalized invariant monotonicity and generalized invexity are established. Our results are generalizations of those presented by Karamardian and Schaible.
128 citations
TL;DR: This paper addresses the problem of finding approximate solutions to MPQP, where the degree of approximation is arbitrary and allows to tradeoff between optimality and a smaller number of cells in the piecewise affine solution.
Abstract: Algorithms for solving multiparametric quadratic programming (MPQP) were recently proposed in Refs. 1–2 for computing explicit receding horizon control (RHC) laws for linear systems subject to linear constraints on input and state variables. The reason for this interest is that the solution to MPQP is a piecewise affine function of the state vector and thus it is easily implementable online. The main drawback of solving MPQP exactly is that, whenever the number of linear constraints involved in the optimization problem increases, the number of polyhedral cells in the piecewise affine partition of the parameter space may increase exponentially. In this paper, we address the problem of finding approximate solutions to MPQP, where the degree of approximation is arbitrary and allows to tradeoff between optimality and a smaller number of cells in the piecewise affine solution. We provide analytic formulas for bounding the errors on the optimal value and the optimizer, and for guaranteeing that the resulting suboptimal RHC law provides closed-loop stability and constraint fulfillment.
128 citations
TL;DR: In this article, the existence of variational-like inequalities with generalized monotone mappings in Banach spaces using the KKM technique was shown. But the results presented in this paper extend and improve the corresponding results of Refs 1-6.
Abstract: In this paper, we introduce two classes of variational-like inequalities with generalized monotone mappings in Banach spaces Using the KKM technique, we obtain the existence of solutions for variational-like inequalities with relaxed η–α monotone mappings in reflexive Banach spaces We present also the solvability of variational-like inequalities with relaxed η–α semimonotone mappings in arbitrary Banach spaces by means of the Kakutani-Fan-Glicksberg fixed-point theorem The results presented in this paper extend and improve the corresponding results of Refs 1–6
118 citations
TL;DR: A new feedback precision-adjustment rule for use with a smoothing technique and standard unconstrained minimization algorithms in the solution of finite minimax problems, which shows that their performance is comparable to or better than that of other algorithms available in the Matlab environment.
Abstract: We present a new feedback precision-adjustment rule for use with a smoothing technique and standard unconstrained minimization algorithms in the solution of finite minimax problems. Initially, the feedback rule keeps a precision parameter low, but allows it to grow as the number of iterations of the resulting algorithm goes to infinity. Consequently, the ill-conditioning usually associated with large precision parameters is considerably reduced, resulting in more efficient solution of finite minimax problems.
104 citations
TL;DR: In this paper, a polynomial-time interior-point algorithm for a class of nonlinear saddle-point problems that involve semidefiniteness constraints on matrix variables is presented.
Abstract: We present a polynomial-time interior-point algorithm for a class of nonlinear saddle-point problems that involve semidefiniteness constraints on matrix variables. These problems originate from robust optimization formulations of convex quadratic programming problems with uncertain input parameters. As an application of our approach, we discuss a robust formulation of the Markowitz portfolio selection model.
TL;DR: In this paper, a systematic transformation of dynamic loads into equivalent static loads has been proposed in Refs. 1-3, where equivalent static forces are made to generate at each time step the same displacement field as the one generated by the dynamic loads.
Abstract: Generally, structural optimization is carried out based on external static loads. However, all forces have dynamic characteristics in the real world. Mathematical optimization with dynamic loads is almost impossible in a large-scale problem. Therefore, in engineering practice, dynamic loads are often transformed into static loads via dynamic factors, design codes, and so on. Recently, a systematic transformation of dynamic loads into equivalent static loads has been proposed in Refs. 1–3. Equivalent static loads are made to generate at each time step the same displacement field as the one generated by the dynamic loads. In this research, it is verified that the solution obtained via the algorithm of Refs. 1–3 satisfies the Karush–Kuhn–Tucker necessary conditions. Application of the algorithm is discussed.
TL;DR: In this article, a dynamic controller for a spacecraft with flexible appendages and based on attitude measurements is proposed, which ensures the asymptotic fulfillment of the objectives in the case of rest-to-rest maneuvers when a failure occurs on the accelerometer sensors, so that the angular velocity is not available for feedback.
Abstract: In this work, we propose a dynamic controller for a spacecraft with flexible appendages and based on attitude measurements. This control ensures the asymptotic fulfillment of the objectives in the case of rest-to-rest maneuvers when a failure occurs on the accelerometer sensors, so that the angular velocity is not available for feedback. Also, it is assumed that the modal variables describing the flexible elements are not measured. This is a lower level controller and is to be selected at the higher level by a supervisor when an emergency situation is detected.
TL;DR: In this article, the existence and continuity of a smooth path for solving the nonlinear complementarity problem with a predictor-corrector smoothing Newton function is discussed. But the authors do not investigate the boundedness of the iteration sequence generated by noninterior continuation/smoothing methods.
Abstract: By smoothing a perturbed minimum function, we propose in this paper a new smoothing function. The existence and continuity of a smooth path for solving the nonlinear complementarity problem (NCP) with a P
0 function are discussed. We investigate the boundedness of the iteration sequence generated by noninterior continuation/smoothing methods under the assumption that the solution set of the NCP is nonempty and bounded. Based on the new smoothing function, we present a predictor-corrector smoothing Newton algorithm for solving the NCP with a P
0 function, which is shown to be globally linearly and locally superlinearly convergent under suitable assumptions. Some preliminary computational results are reported.
TL;DR: In this article, it was shown that the trajectories of a convex function with locally Lipschitz gradient converges weakly to a constrained minimizer whenever it exists.
Abstract: This paper is concerned with the asymptotic analysis of the trajectories of some dynamical systems built upon the gradient projection method in Hilbert spaces. For a convex function with locally Lipschitz gradient, it is proved that the orbits converge weakly to a constrained minimizer whenever it exists. This result remains valid even if the initial condition is chosen out of the feasible set and it can be extended in some sense to quasiconvex functions. An asymptotic control result, involving a Tykhonov-like regularization, shows that the orbits can be forced to converge strongly toward a well-specified minimizer. In the finite-dimensional framework, we study the differential inclusion obtained by replacing the classical gradient by the subdifferential of a continuous convex function. We prove the existence of a solution whose asymptotic properties are the same as in the smooth case.
TL;DR: In this article, the authors consider feedback Nash equilibria in indefinite linear-quadratic differential games on an infinite time horizon, where model uncertainty is represented by a malevolent input which is subject to a cost penalty or to a direct bound.
Abstract: Equilibria in dynamic games are formulated often under the assumption that the players have full knowledge of the dynamics to which they are subject. Here, we formulate equilibria in which players are looking for robustness and take model uncertainty explicitly into account in their decisions. Specifically, we consider feedback Nash equilibria in indefinite linear-quadratic differential games on an infinite time horizon. Model uncertainty is represented by a malevolent input which is subject to a cost penalty or to a direct bound. We derive conditions for the existence of robust equilibria in terms of solutions of sets of algebraic Riccati equations.
TL;DR: In this article, the exact penalty results for nonlinear programs and mathematical programs with equilibrium constraints were proved for strong convexity with order σ, and they were shown to remain valid under some mild conditions.
Abstract: Recently, some exact penalty results for nonlinear programs and mathematical programs with equilibrium constraints were proved by Luo, Pang, and Ralph (Ref. 1). In this paper, we show that those results remain valid under some other mild conditions. One of these conditions, called strong convexity with order σ, is discussed in detail.
TL;DR: In this article, the concept of (S)+ condition for bifunctions is introduced and existence results for equilibrium problems with the (S+) condition are derived. But these results are not applicable to a class of eigenvalue problems.
Abstract: In this paper, we consider equilibrium problems and introduce the concept of (S)+ condition for bifunctions. Existence results for equilibrium problems with the (S)+ condition are derived. As special cases, we obtain several existence results for the generalized nonlinear variational inequality studied by Ding and Tarafdar (Ref. 1) and the generalized variational inequality studied by Cubiotti and Yao (Ref. 2). Finally, applications to a class of eigenvalue problems are given.
TL;DR: A new optimality condition for minimization with general constraints is introduced, which is strictly stronger than and implies the Fritz–John optimality conditions.
Abstract: A new optimality condition for minimization with general constraints is introduced. Unlike the KKT conditions, the new condition is satisfied by local minimizers of nonlinear programming problems, independently of constraint qualifications. The new condition is strictly stronger than and implies the Fritz–John optimality conditions. Sufficiency for convex programming is proved.
TL;DR: Active set strategies for two-dimensional and three-dimensional, unilateral and bilateral obstacle problems are described and efficient computer realizations that are based on multigrid and multilevel methods are suggested.
Abstract: Active set strategies for two-dimensional and three-dimensional, unilateral and bilateral obstacle problems are described. Emphasis is given to algorithms resulting from the augmented Lagrangian (i.e., primal-dual formulation of the discretized obstacle problems), for which convergence and rate of convergence are considered. For the bilateral case, modifications of the basic primal-dual algorithm are also introduced and analyzed. Finally, efficient computer realizations that are based on multigrid and multilevel methods are suggested and different aspects of the proposed techniques are investigated through numerical experiments.
TL;DR: A method for the numerical solution of state-constrained optimal control problems subject to higher-index differential-algebraic equation (DAE) systems is introduced and the projection method guarantees consistency within each iteration, whereas the relaxation method achieves consistency only at an optimal solution.
Abstract: A method for the numerical solution of state-constrained optimal control problems subject to higher-index differential-algebraic equation (DAE) systems is introduced. For a broad and important class of DAE systems (semiexplicit systems with algebraic variables of different index), a direct multiple shooting method is developed. The multiple shooting method is based on the discretization of the optimal control problem and its transformation into a finite-dimensional nonlinear programming problem (NLP). Special attention is turned to the mandatory calculation of consistent initial values at the multiple shooting nodes within the iterative solution process of (NLP). Two different methods are proposed. The projection method guarantees consistency within each iteration, whereas the relaxation method achieves consistency only at an optimal solution. An illustrative example completes this article.
TL;DR: In this paper, a finite-horizon H∞ state-feedback control problem for singularly-perturbed linear time-dependent systems with a small state delay is considered.
Abstract: A finite-horizon H∞ state-feedback control problem for singularly-perturbed linear time-dependent systems with a small state delay is considered. Two approaches to the asymptotic analysis and solution of this problem are proposed. In the first approach, an asymptotic solution of the singularly-perturbed system of functional-differential equations of Riccati type, associated with the original H∞ problem by the sufficient conditions of the existence of its solution, is constructed. Based on this asymptotic solution, conditions for the existence of a solution of the original H∞ problem, independent of the small parameter of singular perturbations, are derived. A simplified controller with parameter-independent gain matrices, solving the original H∞ problem for all sufficiently small values of this parameter, is obtained. In the second approach, the original H∞ problem is decomposed into two lower-dimensional parameter-independent H∞ subproblems, the reduced-order (slow) and the boundary-layer (fast) subproblems; controllers solving these subproblems are constructed. Based on these controllers, a composite controller is derived, which solves the original H∞ problem for all sufficiently small values of the singular perturbation parameter. An illustrative example is presented.
TL;DR: In this paper, a new approach to N-person static fuzzy non-cooperative games and a solution concept such as Nash for these types of games are presented. But it cannot be claimed that the first assumption has been shown to be true in a wide variety of situations involving complex problems in economics, engineering, social and political sciences due to the difficulty inherent in defining an adequate utility function for each player.
Abstract: Systems that involve more than one decision maker are often optimized using the theory of games. In the traditional game theory, it is assumed that each player has a well-defined quantitative utility func- tion over a set of the player decision space. Each player attempts to maximizeminimize hisher own expected utility and each is assumed to know the extensive game in full. At present, it cannot be claimed that the first assumption has been shown to be true in a wide variety of situations involving complex problems in economics, engineering, social and political sciences due to the difficulty inherent in defining an adequate utility function for each player in these types of problems. On the other hand, in many of such complex problems, each player has a heuristic knowledge of the desires of the other players and a heuristic knowledge of the control choices that they will make in order to meet their ends. In this paper, we utilize fuzzy set theory in order to incorporate the players' heuristic knowledge of decision making into the framework of conventional game theory or ordinal game theory. We define a new approach to N-person static fuzzy noncooperative games and develop a solution concept such as Nash for these types of games. We show that this general formulation of fuzzy noncooperative games can be applied to solve multidecision-making problems where no objective function is specified. The computational procedure is illustrated via application to a multiagent optimization problem dealing with the design and oper- ation of future military operations.
TL;DR: In this paper, the authors identify conditions under which time consistency and agreeability, two intertemporal individual rationality concepts, can be verified in linear-state differential games and an illustrative example drawn from environmental economics is provided.
Abstract: The paper identifies conditions under which time consistency and agreeability, two intertemporal individual rationality concepts, can be verified in linear-state differential games. An illustrative example drawn from environmental economics is provided.
TL;DR: In this paper, the authors considered the minimum-time transfer of a satellite from a low and eccentric initial orbit toward a high geostationary orbit, where the thrust available is assumed to be very small (e.g. 0.3 Newton).
Abstract: The minimum-time transfer of a satellite from a low and eccentric initial orbit toward a high geostationary orbit is considered. This study is preliminary to the analysis of similar transfer cases with more complicated performance indexes (maximization of payload, for instance). The orbital inclination of the spacecraft is taken into account (3D model), and the thrust available is assumed to be very small (e.g. 0.3 Newton for an initial mass of 1500 kg). For this reason, many revolutions are required to achieve the transfer and the problem becomes very oscillatory. In order to solve it numerically, an optimal control model is investigated and a homotopic procedure is introduced, namely continuation on the maximum modulus of the thrust: the solution for a given thrust is used to initiate the solution for a lower thrust. Continuous dependence of the value function on the essential bound of the control is first studied. Then, in the framework of parametric optimal control, the question of differentiability of the transfer time with respect to the thrust is addressed: under specific assumptions, the derivative of the value function is given in closed form as a first step toward a better understanding of the relation between the minimum transfer time and the maximum thrust. Numerical results obtained by coupling the continuation technique with a single−shooting procedure are detailed.
TL;DR: The Grover quantum computational search algorithm provides a way to generate the PAS iterates and the resulting implementation, which is called the Grover adaptive search (GAS), realizes PAS for functions satisfying certain conditions, and it is believed that, when quantum computers will be available, GAS will be a practical algorithm.
Abstract: Pure adaptive search (PAS) is an idealized stochastic algorithm for unconstrained global optimization. The number of PAS iterations required to solve a problem increases only linearly in the domain dimension. However, each iteration requires the generation of a random domain point uniformly distributed in the current improving region. If no regularity conditions are known to hold for the objective function, then this task requires a number of classical function evaluations varying inversely with the proportion of the domain constituted by the improving region, entirely counteracting the PAS apparent speedup. The Grover quantum computational search algorithm provides a way to generate the PAS iterates. We show that the resulting implementation, which we call the Grover adaptive search (GAS), realizes PAS for functions satisfying certain conditions, and we believe that, when quantum computers will be available, GAS will be a practical algorithm.
TL;DR: This paper derived a Fan-KKM type theorem and established Fan type geometric properties of convex spaces by applying their results, and also obtained some coincidence theorems and fixed-point theorem in the setting of the convex space.
Abstract: The present paper is divided into two parts In the first part, we derive a Fan-KKM type theorem and establish some Fan type geometric properties of convex spaces By applying our results, we also obtain some coincidence theorems and fixed-point theorems in the setting of convex spaces The second part deals with the applications of our coincidence theorem to establish some existence results for a solution to the generalized vector equilibrium problems
TL;DR: In this paper, the minimum norm solution of a linear program is reformulated as an unconstrained minimization problem with a convex and smooth objective function, and the minimization of this objective function can be carried out by a Newton-type method which is shown to be globally convergent.
Abstract: This paper describes a new technique to find the minimum norm solution of a linear program. The main idea is to reformulate this problem as an unconstrained minimization problem with a convex and smooth objective function. The minimization of this objective function can be carried out by a Newton-type method which is shown to be globally convergent. Furthermore, under certain assumptions, this Newton-type method converges in a finite number of iterations to the minimum norm solution of the underlying linear program.
TL;DR: In this article, a new class of vector quasi-equilibrium problems with set-valued maps is introduced, and a number of C-diagonal quasiconvexity properties are proposed.
Abstract: In this paper, we introduce a new class of vector quasi-equilibrium problems with set-valued maps. Almost all the vector equilibrium models of the Blum-Oettli type in the literature are special cases of our new class of equilibrium problems under consideration. Moreover, a number of C-diagonal quasiconvexity properties are proposed for set-valued maps, which are natural generalizations of the γ-diagonal quasiconvexity for real functions. Together with an application of continuous selection and fixed-point theorems, these conditions enable us to prove unified existence results of solutions for such vector equilibrium problems.
TL;DR: An O(n) time algorithm based on the linear-time median-finding algorithm determines the median of the components of the vector to be projected that was successful in solving all the instances in a reasonable time.
Abstract: We consider the problem of projecting a vector on the intersection of a hyperplane and a box in Rn. This paper extends a previous result of Maculan, Minoux, and Plateau (Ref. 1) concerning the projection of a vector on the intersection of a hyperplane and Rn+. We present an O(n) time algorithm based on the linear-time median-finding algorithm. This algorithm determines the median of the components of the vector to be projected. Computational results are also presented in order to evaluate the algorithm and its time complexity. We consider two sets of instances which are randomly generated for any given n. The algorithm was successful in solving all the instances in a reasonable time.