Optimal Approximation Problem Research Articles

Global impulsive optimal control computation

In this paper, we develop a global computational approach to a classof optimal control problems governed by impulsive dynamical systemsand subject to continuous state inequality constraint. We show thatthis problem is equivalent to an optimal control problem governed byordinary differential equations with periodic boundary conditionsand subject to a set of the continuous state inequality constraints.For this equivalent optimal control problem, a constrainttranscription method is used in conjunction with a penalty functionto construct an appended new cost functional. This leads to asequence of approximate optimal control problems only subject toperiodic boundary conditions. Each of these approximate problems canbe solved as an optimization problem using gradient-basedoptimization techniques. However, these techniques are designed onlyto find local optimal solutions. Thus, a filled function method isintroduced to supplement the gradient-based optimization method.This leads to a combined method for finding a global optimalsolution. A numerical example is solved using the proposed approach.

Aeroelastic Stability Enhancement and Vibration Suppression in a Composite Helicopter Rotor

An optimization procedure to 1) reduce the 4/revolution oscillatory hub loads and 2) increase the lag mode damping of a four-bladed soft-in-plane hingeless helicopter rotor is developed using a two-level approach. At the upper level, response surface approximations to the objective function and constraints are used to find the optimal blade mass and stiffness properties for vibration minimization and stability enhancement. An aeroelastic analysis based on finite elements in space and time is used. The numerical sampling needed to obtain the response surfaces is done using the central composite design of the theory of design of experiments. The approximate optimization problem expressed in terms of quadratic response surfaces is solved using a gradient-based method. Optimization results for the vibration problem in forward flight with unsteady aerodynamic modeling show a vibration reduction of about 15%. The dominant loads are the vertical hub shear and the rolling and pitching moments, which are reduced by 22-26%. The results of stability enhancement problem show an increase of 6-125% in the lag mode damping. At the lower level, a composite box beam is designed to match the upper-level beam blade stiffness and mass using a genetic algorithm which permits the use of discrete ply angle design variables such as 0, +or-45, and 90 deg, which are easier to manufacture. Three different composite materials are used for designing the composite box beam, thus, showing the robustness of the genetic algorithm approach. Boron/epoxy composite gives the most compact box beam, whereas graphite/epoxy gives the lightest box beam

All optimal Hankel-norm approximations and their error bounds in discrete-time†

This paper investigates a parallel problem to Glover (1984): approximate a multivariable transfer function G(z) of McMillan degree n by of McMillan degree smaller than k in discrete-time. A state-space solution is derived to the optimal Hankel-norm approximation problem, together with characterization to all optimal Hankel-norm approximations. It is shown that the error bound derived in Glover (1984) holds for discrete-time systems as well.

Convergence and Approximation of Optimization Problems

In this paper we consider a optimization problem (OP) and study the convergence and approximation of optimal values and optimal solutions to changes in the cost function and the set of feasible solutions. By a general OP we mean that the cost function and the constraints are defined on a Hausdorff topological space. First we obtain convergence results for a OP, and then we present an application of these results on the approximation of the optimal value and the optimal solutions for the so-called capacity problem in metric spaces.

On the Hermitian-Generalized Hamiltonian Solutions of Linear Matrix Equations

By means of the properties of Hermitian-generalized Hamiltonian matrices, we give some necessary and sufficient conditions for the solvability of the matrix equation AX = B in Hermitian-generalized Hamiltonian matrix set $HHC^{n\times n}$. In the case where AX = B is solvable in $HHC^{n\times n}$, we derive the general representation of the solutions. We also give the expression of the solution for the corresponding optimal approximation problem.

Estimation of conditional quantiles by a new smoothing approximation of asymmetric loss functions

In this paper, nonparametric estimation of conditional quantiles of a nonlinear time series model is formulated as a nonsmooth optimization problem involving an asymmetric loss function. This asymmetric loss function is nonsmooth and is of the same structure as the so-called `lopsided' absolute value function. Using an effective smoothing approximation method introduced for this lopsided absolute value function, we obtain a sequence of approximate smooth optimization problems. Some important convergence properties of the approximation are established. Each of these smooth approximate optimization problems is solved by an optimization algorithm based on a sequential quadratic programming approximation with active set strategy. Within the framework of locally linear conditional quantiles, the proposed approach is compared with three other approaches, namely, an approach proposed by Yao and Tong (1996), the Iteratively Reweighted Least Squares method and the Interior-Point method, through some empirical numerical studies using simulated data and the classic lynx pelt series. In particular, the empirical performance of the proposed approach is almost identical with that of the Interior-Point method, both methods being slightly better than the Iteratively Reweighted Least Squares method. The Yao-Tong approach is comparable with the other methods in the ideal cases for the Yao-Tong method, but otherwise it is outperformed by other approaches. An important merit of the proposed approach is that it is conceptually simple and can be readily applied to parametrically nonlinear conditional quantile estimation.

Traffic engineering approach to path selection in optical burst switching networks

It is usually assumed that optical burst switching (OBS) networks use the shortest path routing along with next-hop burst forwarding. The shortest path routing minimizes delay and optimizes utilization of resources, however, it often causes certain links to become congested while others remain underutilized. In a bufferless OBS network in which burst drop probability is the primary metric of interest, the existence of a few highly congested links could lead to unacceptable performance for the entire network. We take a traffic engineering approach to path selection in OBS networks with the objective of balancing the traffic across the network links to reduce congestion and to improve overall performance. We present an approximate integer linear optimization problem as well as a simple integer relaxation heuristic to solve the problem efficiently for large networks. Numerical results demonstrate that our approach is effective in reducing the network-wide burst drop probability, in many cases significantly, over the shortest path routing.

응답량 재사용을 통한 순차 근사최적설계

A general approximate optimization technique by sequential design domain(SDD) did not save response values for getting an approximate functions in each step. It has a disadvantage at aspect of an expense. In this paper, previous response values are recycled for constructing an approximate functions. For this reason, approximation functions is more accurate. Accordingly, even if we did not determine move limit, a system is converged to the optimal design. Size an shape optimization using approximate optimization technique is carried out with SDD. Algorithm executing Pro/Engineer and ANSYS are automatically adopted in the approximate optimization program by SDD. Convergence criterion is defined such that optimal point must be located within SDD during the three steps. The PLBA(Pshenichny-Lim-Belegundu-Arora) algorithm is used to solve approximate optimization problems. This algorithm uses the second-order information in the directions finding problem and used the active set strategy.

Generalized inverse eigenvalue problem for centrohermitian matrices

In this paper we first consider the existence and the general form of solution to the following generalized inverse eigenvalue problem(GIEP): given a set of n-dimension complex vectors {x j } =1 and a set of complex numbers {λ j } =1 , find two n × n centrohermitian matrices A, B such that {x j } =1 and {λj} =1 are the generalized eigenvectors and generalized eigenvalues of Ax=λBx, respectively. We then discuss the optimal approximation problem for the GIEP. More concretely, given two arbitrary matrices, A, $$\tilde B$$ ∈ C n × n , we find two matrices A* and B* such that the matrix (A*, B*) is closest to (A, $$\tilde B$$ ) in the Frobenius norm, where the matrix (A*, B*) is the solution to the GIEP. We show that the expression of the solution of the optimal approximation is unique and derive the expression for it.

Structural optimization of linear dynamical systems under stochastic excitation: a moving reliability database approach

A method to carry out structural synthesis of deterministic linear dynamical systems under stochastic excitation is introduced. The structural optimization problem is written as a nonlinear mathematical programming problem with reliability constraints. Probability that design conditions are satisfied within a given time period is used as a measure of system reliability. The solution of the original optimization problem is replaced by the solution of a sequence of approximate sub-optimization problems. An explicit approximation of the system reliability in terms of the design variables is constructed in each sub-optimization problem. The approximations are locally adjusted to a reliability database, which is obtained by an efficient importance sampling technique. Each approximate optimization problem is solved in an efficient manner due to the availability of the system reliability in explicit form. Numerical examples are presented to illustrate the performance and efficiency of the proposed methodology.

Formula omitted] additive and [formula omitted] ultra-additive maps, Gromov's trees, and the Farris transform

In phylogenetic analysis, one searches for phylogenetic trees that reflect observed similarity between a collection of species in question. To this end, one often invokes two simple facts: (i) Any tree is completely determined by the metric it induces on its leaves (which represent the species). (ii) The resulting metrics are characterized by their property of being additive or, in the case of dated rooted trees, ultra-additive. Consequently, searching for additive or ultra-additive metrics A that best approximate the metric D encoding the observed similarities is a standard task in phylogenetic analysis. Remarkably, while there are efficient algorithms for constructing optimal ultra-additive approximations, the problem of finding optimal additive approximations in the l 1 or l ∞ sense is NP-hard. In the context of the theory of δ - hyperbolic groups, however, good additive approximations A of a metric D were found by Gromov already in 1988 and shown to satisfy the bound ∥ D - A ∥ ∞ ⩽ Δ ( D ) ⌈ log 2 ( # X - 1 ) ⌉ , where Δ ( D ) , the hyperbolicity of D, i.e. the maximum of all expressions of the form D ( u , v ) + D ( x , y ) - max ( D ( u , x ) + D ( v , y ) , D ( u , y ) + D ( v , x ) ) ( u , v , x , y ∈ X ) . Yet, besides some notable exceptions (e.g. Adv. Appl. Math. 27 (2001) 733–767), the potential of Gromov's concept of hyperbolicity is far from being fully explored within the context of phylogenetic analysis. In this paper, we provide the basis for a systematic theory of Δ ultra-additive and Δ additive approximations. In addition, we also explore the average and worst case behavior of Gromov's bound.

The solvability conditions for the inverse eigenvalue problem of the symmetrizable matrices

The necessary and sufficient conditions for the solvability of the inverse eigenvalue problem AX= XΛ over the class of the symmetrizable matrices are discussed, the general expression of the solution is given. Moreover, the related optimal approximation problem to a given matrix over the solution set is solved.

Galerkin method for optimal control of second‐order evolution equations

AbstractThis paper presents the Galerkin approximation of the optimization problem of a system governed by non‐linear second‐order evolution equation where a non‐linear operator depends on derivative of the state of the system. The control is acting on a non‐linear equation. After giving some results on the existence of optimal control we shall prove the existence of the weak and strong condensation points of a set of solutions of the approximate optimization problems. Each of these points is a solution of the initial optimization problem. Finally we shall give a simple example using the obtained results. Copyright © 2004 John Wiley & Sons, Ltd.

Existence Theorems and Galerkin Approximation for Non-Linear Evolution Control Problems

This article presents an optimization problem involving a system governed by a non-linear parabolic equation (dy/dt) + Ay = u where A is a radially continuous, monotone and coercive Volterra operator with the cost functional weakly lower semi-continuous and radially unbounded (coercive). In a first preparatory part of the article we prove two existence theorems. In the second part we present the Galerkin approximation and we prove existence of the weak and strong condensation points of a set of solution of the approximate optimization problems. Each of this points is a solution of the initial optimization problem. Finally we shall give examples using the obtained results.

Maximum Principle of Optimal Control for Degenerate Quasi-Linear Elliptic Equations

Optimal control problems governed by degenerate quasi-linear partial differential equations of elliptic type are considered. The optimal control systems considered may lack Cesari-type conditions, and therefore the corresponding approximate optimal control problem may have no solution. To yield the maximum principle of optimal pairs, relaxed controls are used to overcome the difficulties occurring when considering approximate problems. The relaxed controls used are defined by finite additive measures so that the case of the control set being noncompact can be treated.

An Approach to the Optimization and Approximation of Solutions of any Operator Inclusion

In this paper we shall present the optimization problem in a set of solutions of operator inclusion with the maximal monotone operator. Then we treat optimization problem by Galerkin method and we prove convergence of optimal values of approximated optimization problems to the one for the original problem. Finally, we apply the results to give a simple example.

Dynamic optimization of dissipative PDE systems using nonlinear order reduction

This article presents computationally efficient methods for the solution of dynamic constraint optimization problems arising in the context of spatially distributed processes governed by highly dissipative nonlinear partial differential equations (PDEs). The methods are based on spatial discretization using the method of weighted residuals with spatially global basis functions (i.e., functions that cover the entire domain of definition of the process and satisfy the boundary conditions). More specifically, we perform spatial discretization of the optimization problems using the method of weighted residuals with analytical or empirical (obtained via Karhunen–Loève expansion) eigenfunctions as basis functions, and combination of the method of weighted residuals with approximate inertial manifolds. The proposed methods account for the fact that the dominant dynamics of highly dissipative PDE systems are low dimensional in nature and lead to approximate optimization problems that are of significantly lower order compared to the ones obtained from spatial discretization using finite-difference and finite-element techniques, and thus, they can be solved with significantly smaller computational demand. The resulting dynamic nonlinear programs include equality constraints that constitute a low-order system of coupled ordinary differential equations and algebraic equations, which can then be solved with combination of standard temporal discretization and nonlinear programming techniques. We employ backward finite differences (implicit Euler) to perform temporal discretization and solve the nonlinear programs resulting from temporal and spatial discretization using reduced gradient techniques (MINOS). We use two representative examples of dissipative PDEs, a diffusion-reaction process with constant and spatially varying coefficients, and the Kuramoto–Sivashinsky equation, a model that describes incipient instabilities in a variety of physical and chemical systems, to demonstrate the implementation and evaluate the effectiveness of the proposed optimization algorithms.

The solvability conditions for inverse eigenproblem of symmetric and anti‐persymmetric matrices and its approximation

AbstractThe problem of generating a matrix A with specified eigen‐pair, where A is a symmetric and anti‐persymmetric matrix, is presented. An existence theorem is given and proved. A general expression of such a matrix is provided. We denote the set of such matrices by 𝒮𝒜𝒮En. The optimal approximation problem associated with 𝒮𝒜𝒮En is discussed, that is: to find the nearest matrix to a given matrix A* by A∈𝒮𝒜𝒮En. The existence and uniqueness of the optimal approximation problem is proved and the expression is provided for this nearest matrix. Copyright © 2002 John Wiley & Sons, Ltd.

The solvability conditions for the inverse eigenvalue problem of Hermitian-generalizedHamiltonian matrices

In this paper, the inverse eigenvalue problem of Hermitian-generalizedHamiltonian matrices and their optimal approximation problem are considered.Some sufficient and necessary conditions of the solvability for the inverseeigenvalue problem are given. A general representation of the solution is presentedfor a solvable case. Furthermore, an expression of the solution for the optimalapproximation problem is presented.

Numerical solution of an optimal control problem with variable time points in the objective function

AbstractIn this paper, we consider the numerical solution of a class of optimal control problems involving variable time points in their cost functions. The control enhancing transform is first used to convert the optimal control problem with variable time points into an equivalent optimal control problem with fixed multiple characteristic time (MCT). Using the control parametrization technique, the time horizon is partitioned into several subintervals. Let the partition points also be taken as decision variables. The control functions are approximated by piecewise constant or piecewise linear functions in accordance with these variable partition points. We thus obtain a finite dimensional optimization problem. The control parametrization enhancing control transform (CPET) is again used to convert approximate optimal control problems with variable partition points into equivalent standard optimal control problems with MCT, where the control functions are piecewise constant or piecewise linear functions with pre-fixed partition points. The transformed problems are essentially optimal parameter selection problems with MCT. The gradient formulae for the objective function as well as the constraint functions with respect to relevant decision variables are obtained. Numerical examples are solved using the proposed method.