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This paper deals with a fully implicit time discretization scheme with variable time-step for a nonlinear system modelling phase transition and mechanical deformations in shape memory alloys. The model is studied in the non-stationary case and accounts for local microscopic interactions between the phases introducing the gradients of the phase parameters. The resulting initial-boundary value problem has already been studied by the author who proved existence, uniqueness and continuous dependence on data for a suitable weak solution along some regularity results. A careful and detailed investigation of the variable time-step discretization is the goal of this paper. Thus, we deduce some estimates for the discretization error. These estimates depend only on data, impose no constraints between consecutive time-steps and show an optimal order of convergence. Finally, we prove another regularity result for the solution under stronger regularity assumptions on data. [ABSTRACT FROM AUTHOR]

In this paper, we propose a decentralized parallel computation model for global optimization using interval analysis. The model is adaptive to any number of processors and the workload is automatically and evenly distributed among all processors by alternative message passing. The problems received by each processor are processed based on their local dominance properties, which avoids unnecessary interval evaluations. Further, the problem is treated as a whole at the beginning of computation so that no initial decomposition scheme is required. Numerical experiments indicate that the model works well and is stable with different number of parallel processors, distributes the load evenly among the processors, and provides an impressive speedup, especially when the problem is time-consuming to solve. [ABSTRACT FROM AUTHOR]

In a 1966 paper by C.A.B. Smith, the short and long conjunctive sums of games are defined and methods are described for determining the theoretical winner of a game constructed using one type of these sums. In this paper, we develop a method for determining the winner of a game constructed using arbitrary combinations of these sums. [ABSTRACT FROM AUTHOR]

A novel filled function is given in this paper to find a global minima for a nonsmooth constrained optimization problem. First, a modified concept of the filled function for nonsmooth constrained global optimization is introduced, and a filled function, which makes use of the idea of the filled function for unconstrained optimization and penalty function for constrained optimization, is proposed. Then, a solution algorithm based on the proposed filled function is developed. At last, some preliminary numerical results are reported. The results show that the proposed approach is promising. [ABSTRACT FROM AUTHOR]

In this paper, we study one kind of stochastic recursive optimal control problem with the obstacle constraint for the cost functional described by the solution of a reflected backward stochastic differential equation. We give the dynamic programming principle for this kind of optimal control problem and show that the value function is the unique viscosity solution of the obstacle problem for the corresponding Hamilton–Jacobi–Bellman equation. [ABSTRACT FROM AUTHOR]

The combination of geometrically nonlinear analysis with structural optimization against instability opens many possibilities for new formulations of the optimization problem. One of them is optimization against instability in the large. When a critical state does not exist and instability occurs at finite displacements either the lower critical loading is maximized or the generalized displacement for that load is minimized. This paper undertakes these new optimization problems providing formulations and numerical solutions for selected elements. The simple finite dimensional model is analysed first showing characteristic features of the problem, and then a real elastic element is optimized. [ABSTRACT FROM AUTHOR]

In the paper, a new trust region algorithm is given for nonlinear inequality constrained optimization problems. Motivated by a dual problem introduced by Han and Mangasarian [Han, S. P., Mangasarian, O. L. (1983). A dual differentiable exact penalty function. Math. Programming 25:293-306], which is a nonnegatively constrained maximization problem, we construct a trust region algorithm for solving the dual problem. At each iteration, we only need to minimize a quadratic subproblem with simple bound constraints. Under the condition that the iterate sequence generated by the algorithm is contained in some bounded closed set, any accumulation point of the sequence is a Karush- Kuhn-Tucker point of the original problem. [ABSTRACT FROM AUTHOR]

The aim of this paper is to generalize the results of Yamada et al. [Yamada, I., Ogura, N., Yamashita, Y., Sakaniwa, K. (1998). Quadratic approximation of fixed points of nonexpansive mappings in Hilbert spaces. Numer. Funct. Anal. Optimiz. 19(l):165-190], and to provide complementary results to those of Deutsch and Yamada [Deutsch, F., Yamada, I. (1998). Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings. Numer. Funct. Anal. Optim. 19(1&2):33-56] in which they consider the minimization of some function d over a closed convex set F, the nonempty intersection of N fixed point sets. We start by considering a quadratic function θ and providing a relaxation of conditions of Theorem 1 of Yamada et al. (1998) to obtain a sequence of fixed points of certain contraction maps, converging to the unique minimizer of θ over F. We then extend Theorem 2 and obtain a complementary result to Theorem 3 of Yamada et al. (1998) by replacing the condition lim n → ∞ (λn - λn+1)/λ²n+1 = 0 on the parameters by the more general condition lim n → ∞ λn / λn+1 = 1. We next look at minimizing a more general function θ than a quadratic function which was proposed by Deutsch and Yamada (1998) and show that the sequence of fixed points of certain maps converge to the unique minimizer of 9 over F. Finally, we prove a complementary result to that of Deutsch and Yamada (1998) by using the alternate condition on the parameters. [ABSTRACT FROM AUTHOR]

This paper deals with the numerical solution of optimal control problems for ODEs. The methods considered here rely on some standard optimization code to solve a discretized version of the control problem under consideration. We aim to make available to the optimization software not only the discrete objective functional, but also its gradient. The objective gradient can be computed either from forward (sensitivity) information or backward (adjoint) information. The purpose of this paper is to discuss various ways of adjoint computation. It will be shown both theoretically and numerically that methods based on the continuous adjoint equation require a careful choice of both the integrator and gradient assembly formulas in order to obtain a gradient consistent with the discretized control problem. Particular attention is given to automatic differentiation techniques which generate automatically a suitable integrator. [ABSTRACT FROM AUTHOR]

The Fermat-Torricelli problem is an optimization problem associated with a finite subset $\{a_{j}\}_{j=1}^{q}$ of ℝ and a family $\{c_{j}\}_{j=1}^{q}$ of positive weights. The function F to be minimized is defined by $F(x)=\sum _{j=1}^{q}c_{j}\Vert x-a_{j}\Vert$. In this paper, we extend this problem to the case of volumes. [ABSTRACT FROM AUTHOR]

The notion of extended-well-posedness has been introduced by Zolezzi for scalar minimization problems and has been further generalized to vector minimization problems by Huang. In this paper, we study the extended well-posedness properties of vector minimization problems in which the objective function is C-quasiconvex. To achieve this task, we first study some stability properties of such problems. [ABSTRACT FROM AUTHOR]

In this paper we present sufficient conditions for global optimality of non-convex quadratic programs involving linear matrix inequality (LMI) constraints. Our approach makes use of the concept of a quadratic subgradient. We develop optimality conditions for quadratic programs with LMI constraints by using Lagrangian function and by examining conditions which minimizes a quadratic subgradient of the Lagrangian function over simple bounding constraints. As applications, we obtain sufficient optimality condition for quadratic programs with LMI and box constraints by minimizing a quadrtic subgradient over box constraints. We also give optimality conditions for quadratic minimization involving LMI and binary constraints. [ABSTRACT FROM AUTHOR]

The idea of containment sets (csets) is due to Walster and Hansen, and the theory is mainly due to the first author. Now that floating point computation with infinities is widely accepted, it is necessary to achieve the same for interval computation. The cset of a function over a set in its domain space is the set of all limits of normal function values over that set. Csets form a sound basis for defining a number of practical models for interval arithmetic that handle division by zero and related operations in an intuitive and consistent manner. Cset-based systems offer new opportunities for compiler optimization by rearranging interval expressions, achieving tighter bounds by reducing dependencies within the expression. This paper presents basic theory. It discusses division by zero, the case for a global flag to support ``loose'' evaluation, performance, and semantics. It presents numerical examples using a trial Matlab implementation. [ABSTRACT FROM AUTHOR]

In the procedure of steady-state hierarchical optimization for large-scale industrial processes, it is often necessary that the control system responds to a sequence of step function-type control decisions with distinct magnitudes. In this paper a set of iterative learning controllers are de-centrally embedded into the procedure of the steady-state optimization. This generates upgraded sequential control signals and thus improves the transient performance of the discrete-time large-scale systems. The convergence of the updating law is derived while the intervention from the distinction of the scales is analysed. Further, an optimal iterative learning control scheme is also deduced by means of a functional derivation. The effectiveness of the proposed scheme and the optimal rule is verified by simulation. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the smoothing of an implicit function defined by a nonsmooth underdetermined system of equations F(y, z) =0. We apply a class of parametrized smoothing methods to smooth F and investigate the limiting behavior of the implicit function solving the smoothed equations. In particular, we discuss the approximation of the Clarke generalized Jacobian of the implicit function when F is piecewise smooth. As an application, we present an analysis of the generalized Karush-Kuhn-Tucker conditions of different forms for a piecewise-smooth equality-constrained minimization problem. [ABSTRACT FROM AUTHOR]

In this paper, we consider a general primal-dual nonlinear rescaling (PDNR) method for convex optimization with inequality constraints. We prove the global convergence of the PDNR method and estimate the error bounds for the primal and dual sequences. In particular, we prove that, under the standard second-order optimality conditions, the error bounds for the primal and dual sequences converge to zero with linear rate. Moreover, for any given ratio 0 < γ < 1, there is a fixed scaling parameter kγ > 0 such that each PDNR step shrinks the primal-dual error bound by at least a factor 0 < γ < 1, for any k ≥ kγ. The PDNR solver was tested on a variety of NLP problems including the constrained optimization problems (COPS) set. The results obtained show that the PDNR solver is numerically stable and produces results with high accuracy. Moreover, for most of the problems solved, the number of Newton steps is practically independent of the problem size. [ABSTRACT FROM AUTHOR]

The trust region subproblem plays an important role in optimization and numerical analysis. Many researchers even use it in regularizing ill-posed problems. It appears that the trust region subproblem is ill-posed: the set of solutions is unstable with respect to the data in the functional to be minimized, that is a small error in the functional to be minimized might cause large errors in the set of solutions. The aim of the paper is to study the ill-posed nature of the problem and to suggest methods to overcome the ill-posedness. The methods are mainly based on Tikhonov regularization with the generalized discrepancy principle suggested by Goncharskii, Leonov, and Yagola and the difference of convex functions algorithm (DCA) recently developed by Plant Dinh Tao and Le Thi Hoai An. The open problem of Tikhonov regularization methods for non-linear ill-posed problems how to globally solve non-linear (in general non-convex) optimization problems occurred from them is completely answered for the trust region subproblem by DCA. Several test numerical examples are outlined. [ABSTRACT FROM AUTHOR]

This paper is an extension of previously published work by Bernard O. Koopman to the general case of any number of tasks and any effect function. A systematic method for the numerical calculation of the maximum effect and its corresponding optimum distribution is presented in a form which is also suitable for solution on an automatic digital computer. For the special case of additive returns which saturate, a special graphical or numerical method for any number of tasks is presented with an example worked out in detail A similar method for multiplicative effects is sketched. [ABSTRACT FROM AUTHOR]

In multiple-hop MIMO relay systems, an end-to-end channel capacity is restricted by the bottleneck relay. Therefore, in order to obtain the high end-to-end channel capacity, we propose a simple mathematical method to optimize both distances and transmit powers simultaneously. Additionally, the end-to-end channel capacity of optimization based on an one-way and a two-ways transmissions is analyzed. The specific TDMA is proposed to control the transmission of all transmitters on MAC layer and then the distance and the transmit power are optimized based on MAC-PHY cross layer by the proposal particle filter method to obtain the higher end-to-end channel capacity. The calculation result indicates that there is the optimal number of relays that has the maximal end-to-end channel capacity and the trade-off between the end-to-end channel capacity and the delay time. [ABSTRACT FROM AUTHOR]

This work presents an algorithm for multiobjective optimization that is structured as: (i) a descent direction is calculated, within the cone of descent and feasible directions, and (ii) a multiobjective line search is conducted over such direction, with a new multiobjective golden section segment partitioning scheme that directly finds line-constrained efficient points that dominate the current one. This multiobjective line search procedure exploits the structure of the line-constrained efficient set, presenting a faster compression rate of the search segment than single-objective golden section line search. The proposed multiobjective optimization algorithm converges to points that satisfy the Kuhn-Tucker first-order necessary conditions for efficiency (the Pareto-critical points). Numerical results on two antenna design problems support the conclusion that the proposed method can solve robustly difficult nonlinear multiobjective problems defined in terms of computationally expensive black-box objective functions. [ABSTRACT FROM AUTHOR]

In this paper, a new conjugate gradient formula βNewk is given to compute the search directions for unconstrained optimization problems. General convergence results for the proposed formula with some line searches such as the exact line search, the Grippo-Lucidi line search and the Wolfe-Powell line search are discussed. Under the above line searches and some assumptions, the global convergence properties of the given methods are discussed. The given formula βNewk ⩾ 0, and the search directions dk which are generated by the given βNewk under the strong Wolfe-Powell line search satisfy the sufficient descent condition. Preliminary numerical results show that the proposed methods are efficient. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider a class of non-standard time optimal control problems involving a dynamical system consisting of multiple subsystems evolving over different time horizons. Different subsystems are required to reach their respective target sets at different termination times. The goal is to minimize the maximum of these termination times. By introducing a discrete variable to represent the system termination ordering, we reformulate this problem as a discrete optimization problem. A discrete filled function method is developed to solve this discrete optimization problem. For illustration, a numerical example is solved. [Copyright &y& Elsevier]

Abstract: The shape optimization problem consists in looking for the geometry that minimizes an objective function, like mass or compliance, subject to mechanical constraints. The boundary element method (BEM) is used for the structural analysis. For linear elasticity problems, it needs only a mesh on the boundary of the structure. This characteristic makes the BEM a natural method for shape optimization, since only the boundary is needed to define the optimization problem and to carry out the structural analysis. The simultaneous analysis and design formulation (SAND) for structural optimization considers the state variables as unknowns of the optimization problem and includes the equilibrium equations as equality constraints. In this way, it is not necessary to solve the equilibrium equation per iteration; the equilibrium is only obtained at the end of the optimization process. In this paper, the shape optimization problem is dealt with using the BEM formulation to define a SAND optimization problem that is solved using an interior point algorithm. Numerical results for two-dimensional linear elasticity problems are presented to illustrate the efficacy of the proposed technique. [Copyright &y& Elsevier]

We present a new constraint qualification which guarantees strong duality between a cone-constrained convex optimization problem and its Fenchel-Lagrange dual. This result is applied to a convex optimization problem having, for a given nonempty convex cone K, as objective function a K-convex function postcomposed with a K-increasing convex function. For this so-called composed convex optimization problem, we present a strong duality assertion, too, under weaker conditions than the ones considered so far. As an application, we rediscover the formula of the conjugate of a postcomposition with a K-increasing convex function as valid under weaker conditions than usually used in the literature. [ABSTRACT FROM AUTHOR]

Object-oriented programming is a relatively new tool in the development of optimization software. The code extensibility and the rapid algorithm prototyping capability enabled by this programming paradigm promise to enhance the reliability, utility, and ease of use of optimization software. While the use of object-oriented programming is growing, there are still few examples of general purpose codes written in this manner, and a common approach is far from obvious. This paper describes OPT++, a C++ class library for nonlinear optimization. The design is predicated on the concept of distinguishing between an algorithm-independent class hierarchy for nonlinear optimization problems and a class hierarchy for nonlinear optimization methods that is based on common algorithmic traits. The interface is designed for ease of use while being general enough so that new optimization algorithms can be added easily to the existing framework. A number of nonlinear optimization algorithms have been implemented in OPT++ and are accessible through this interface. Furthermore, example applications demonstrate the simplicity of the interface as well as the advantages of a common interface in comparing multiple algorithms. [ABSTRACT FROM AUTHOR]

A new approach to a solution of a nonlinear constrained mathematical programming problem and its Mond-Weir duals is introduced. An n-approximated problem associated with a primal nonlinear programming problem is presented that involves n-approximated functions constituting the primal problem. The equivalence between the original mathematical programming problem and its associated n-approximated optimization problem is established under invexity assumption. Furthermore, n-approximated dual problems in the sense of Mond-Weir are introduced for the obtained n-approximated optimization problem in this method. By the help of n-approximated dual problems some duality results are established for the original mathematical programming problem and its original duals. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of optimizing a function over a finite or countably infinite set of alternatives, in situations where this objective function cannot be evaluated exactly, but has to be estimated or measured. A special focus is on situations where simulation is used to evaluate the objective function. We present two versions of a new iterative method for solving such discrete stochastic optimization problems. In each iteration of the proposed method, a neighbor of the "current" alternative is selected, and estimates of the objective function evaluated at the current and neighboring alternatives are compared. The alternative that has a better observed function value becomes the next current alternative. We show how one version of the proposed method can be used to solve discrete optimization problems where the objective function is evaluated using transient or steady-state simulation, and we show how the other version can be applied to solve a special class of discrete stochastic optimization problems and present some numerical results. A major strength of the proposed method is that it spends most of the computational effort at local minimizers of the objective function. In fact, we show that for both versions of the proposed method, the alternative that has been visited most often in the first m iterations converges almost surely to a local optimizer of the objective function as m goes to infinity. [ABSTRACT FROM AUTHOR]

Explains that this special issue of "Numerical Linear Algebra with Applications" is devoted to the 2002 International Conference on "Preconditioning Methods for Optimal Control and Constrained Optimization Problems" held on October 23-25, 2002 in Netherlands. Goal of addressing complex issues related to preconditioning methods for constrained optimization and optimal control.

The article discusses a study on a class of correct mathematical operations on prediction and recognition algorithms. It states that the right operations come up in relation to the issue of constructing correct algorithms and that correct operations possess the property of preserving the correctness of the algorithms. It adds that the property is useful in developing procedures of correct training that involves the potential of generating families of correct algorithms. The application of a correct operation reportedly makes it possible to obtain new algorithms in an extended model of algorithms.

A hyperplane search tree is a binary tree used to store a set S of n d-dimensional data points. In a random hyperplane search tree for S, the root represents a hyperplane defined by d data points drawn uniformly at random from S. The remaining data points are split by the hyperplane, and the definition is used recursively on each subset. We assume that the data are points in general position in Rd. We show that, uniformly over all such data sets S, the expected height of the hyperplane tree is not worse than that of the k-d tree or the ordinary one-dimensional random binary search tree, and that, for any fixed d ≥ 3, the expected height improves over that of the standard random binary search tree by an asymptotic factor strictly greater than one. [ABSTRACT FROM AUTHOR]

A class of optimal control problems for a semilinear elliptic partial differential equation with control constraints is considered. It is well known that sufficient second-order conditions ensure the stability of optimal solutions, and the convergence of numerical methods. Otherwise, such conditions are very difficult to verify (analytically or numerically). We will propose a new approach as follows: Starting with a numerical solution for a fixed mesh we will show the existence of a local minimizer of the continuous problem. Moreover, we will prove that this minimizer satisfies the sufficient second-order conditions. [ABSTRACT FROM AUTHOR]

By refining a variant of the Klee–Minty example that forces the central path to visit all the vertices of the Klee–Minty n-cube, we exhibit a nearly worst-case example for path-following interior point methods. Namely, while the theoretical iteration-complexity upper bound is $$O(2^{n}n^{\frac{5}{2}})$$ , we prove that solving this n-dimensional linear optimization problem requires at least 2 n −1 iterations. [ABSTRACT FROM AUTHOR]

A mixed optimal model approximation is presented to obtain reduced-order models for truly fast descriptor systems. By a projection from truly fast descriptor systems to discrete-time systems, a mixed optimal model approximation for truly fast descriptor systems is transformed to a mixed optimal model approximation of the corresponding discrete-time systems. The structure of the fast descriptor systems is preserved in the model approximation procedure. The expression of the error and its gradient are given explicitly in terms of the solutions of certain Lyapunov equations. A numerical example is provided to illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this article, we propose a method to identify contamination events (location and time of release) by enhancing a mathematical method originally proposed by Carasso et al. (Carasso, A., Sanderson, J.G. and Hyman, J.M., 1978, Digital removal of random media image degradation by solving the diffusion equation backward in time. SIAM Journal of Numerical Analysis, 15(4)). The method of the Marching-Jury Backward Beam/Plate Equation, applied earlier to groundwater problems, is enhanced and coupled to discrete Fourier transform processing techniques to solve a two-dimensional (2D) advection-dispersion transport problem with homogeneous and isotropic parameters backwards in time. (Atmadja, J. and Bagtzoglou, A.C., 2001a, Pollution source identification in heterogeneous porous media. Water Resources Research, 37(8), 2113–2125; Bagtzoglou, A.C. and Atmadja, J., 2003, The marching-jury backward beam equation and quasi-reversibility methods for hydrologic inversion: application to contaminant plume spatial distribution recovery. Water Resources Research, 39(2), 1038. Cornacchiulo, D. and Bagtzoglou, A.C., 2002, The marching-jury backward plate equation for contaminant plume spatial distribution recovery in two-dimensional heterogeneous media: Computational Issues, In: S.M. Hassanizadeh, R.J. Schotting, W.G. Gray and G.F. Pinder (Eds.) Computational Methods for Subsurface Flow and Transport (Netherlands: Elsevier Publishers) pp. 461–468. The difficulties associated with this ill-posed, inverse problem are well-recognized (Atmadja, J. and Bagtzoglou, A.C., 2001b, State-of-the-art report on mathematical methods for groundwater pollution source identification. Environmental Forensics, 2(3), 205–214). We, therefore, enhance the method by integrating an optimization scheme that takes as input parameters the stabilization parameter, the transport velocities, and the coefficient of diffusion. The objective function is set as an equally weighted sum of different mass and peak errors that can be calculated based on a combination of exhaustive contaminant coverage at specific points in time (e.g., lidar) and/or point data collected at a continuously monitored network of chemical sensors or biosensors, which may be stationary or mobile. [ABSTRACT FROM AUTHOR]

A chaotic algorithm for providing a solution to the bi-level Continuous Equilibrium Network Design Problem (NDP) is discussed following an introduction of the Continuous Network Design Problem (GNDP) model with road reserve and Chaos Optimization Algorithms (COA). A description of the current optimality conditions for the CNDP model is described in details. Then the chaotic approach for the GNDP is applied to two numerical examples. The results have been encouraging, indicating that the chaotic approach has great potential ability in finding the optimal solution of CNDP models. [ABSTRACT FROM AUTHOR]

The aim of this article is to study necessary optimality conditions for a vector minimization program involving locally Lipschitz functions under certain general regularity conditions. We study problems involving only inequality and both inequality and equality constraints. [ABSTRACT FROM AUTHOR]

We develop an implementable algorithm for stochastic optimization problems involving probability functions. Such problems arise in the design of structural and mechanical systems. The algorithm consists of a nonlinear optimization algorithm applied to sample average approximations and a precision-adjustment rule. The sample average approximations are constructed using Monte Carlo simulations or importance sampling techniques. We prove that the algorithm converges to a solution with probability one and illustrate its use by an example involving a reliability-based optimal design. [ABSTRACT FROM AUTHOR]

Sufficient conditions are derived for the controllability of nonlinear neutral evolution integrodifferential systems with infinite delay in a Banach space. The results are obtained by using the resolvent operators and the Schaefer fixed-point theorem. An example is given to illustrate the results. [ABSTRACT FROM AUTHOR]