Optimal Approximation Problem Research Articles

Optimal discrete-valued control computation

In this paper, we consider an optimal control problem in which the control takes values from a discrete set and the state and control are subject to continuous inequality constraints. By introducing auxiliary controls and applying a time-scaling transformation, we transform this optimal control problem into an equivalent problem subject to additional linear and quadratic constraints. The feasible region defined by these additional constraints is disconnected, and thus standard optimization methods struggle to handle these constraints. We introduce a novel exact penalty function to penalize constraint violations, and then append this penalty function to the objective. This leads to an approximate optimal control problem that can be solved using standard software packages such as MISER. Convergence results show that when the penalty parameter is sufficiently large, any local solution of the approximate problem is also a local solution of the original problem. We conclude the paper with some numerical results for two difficult train control problems.

On the η − (1, 2) approximated optimization problems

Let X be a nonempty subset of Rn, x 0 be an interior point of X, f : X → R be a differentiable function at x 0 , g : X → Rm be a twice differentiable function at x 0 and η : X × X → Rn be a function. In this paper, we attach to the optimization problem ... the (1, 2)-η- approximated optimization problem ... and we will study the relations between the optimal solutions of Problem (P), the optimal solutions of Problem (AP), the saddle points of Problem (P) and saddle points of Problem (AP).

Optimization problems, first order approximated optimization problems and their connections

In this paper, we attach to the optimization problem ... where X is a subset of Rn, f : X → R, g = (g1, ..., gm) : X → Rm and h = (h1, ..., hq) : X → Rq are functions, the (0, 1) − η− approximated optimization problem (AP). We will study the connections between the optimal solutions for Problem (AP), the saddle points for Problem (AP), optimal solutions for Problem (P) and saddle points for Problem (P).

Solution to a Class of Matrix Equations with k-Involutary Symmetrices

In this paper, we investigate the solvability of matrix equations with -involutary symmetric matrix , the general solution of which is obtained when it is solvable. Meantime, the associated optimal approximation problem for some given matrix is also considered under some particular hypothesis.

On the Hermitian R‐Conjugate Solution of a System of Matrix Equations

Let R be an n by n nontrivial real symmetric involution matrix, that is, R = R−1 = RT ≠ In. An n × n complex matrix A is termed R‐conjugate if , where denotes the conjugate of A. We give necessary and sufficient conditions for the existence of the Hermitian R‐conjugate solution to the system of complex matrix equations AX = C and XB = D and present an expression of the Hermitian R‐conjugate solution to this system when the solvability conditions are satisfied. In addition, the solution to an optimal approximation problem is obtained. Furthermore, the least squares Hermitian R‐conjugate solution with the least norm to this system mentioned above is considered. The representation of such solution is also derived. Finally, an algorithm and numerical examples are given.

The Hermitian Reflexive Solutions and The Anti-Hermitian Reflexive Solutions for A Class of Matrix Equations (AX=B, XC=D)

In this paper, The Hermitian reflexive solutions and the anti-Hermitian reflexive solutions of matrix equations AX = B, XC = D are considered. With special properties of partitioned matrices and Hermitian reflexive (anti-Hermitian reflexive) matrices, the necessary and sufficient conditions for the solvability and the general expression of the solution are obtained. Moreover, the related optimal approximation problem to a given matrix over the solution set is considered.

Spacecraft long-duration phasing maneuver optimization using hybrid approach

Before a manned orbital rendezvous mission, the target spacecraft usually performs several maneuvers to adjust the initial phase angle of the orbital rendezvous and to coordinate the injection of the chaser. This maneuvering process is referred to as the “target phasing mission”. This target phasing presents an orbital long-duration two-point boundary value problem. Further, when the maneuver revolution numbers are used as design variables, the target phasing maneuver's optimization becomes a mixed integer nonlinear programming problem. This paper presents a new optimization method for this phasing maneuver mission, employing a hybrid approach. First, we provide an approximate phasing optimization problem that considers the phase angle influences of node drift and orbital altitude decay. This problem is then optimized using a hybrid approach that integrates branch-and-bound and sequential quadratic programming. Second, a shooting iteration method is adopted to improve the solution to the approximate problem in order to satisfy the terminal constraints of high-precision numerical integration. The proposed method is then applied to an operational target phasing maneuver problem. The results lead to four major conclusions: (1) The proposed approximate phasing optimization model presents a good approximation of the operational mission. (2) The hybrid optimization approach can solve the approximate problem effectively, and the shooting iteration used to arrive at a high-precision solution converges steadily and rapidly. (3) Compared with mixed-code genetic algorithm, the proposed method can obtain a similar result with a lower computation cost and, compared with the approximate model that does not consider node drift and orbital altitude decay, the proposed method has better convergence efficiency. (4) The terminal time of target phasing remains almost constant when the initial semi-major axis increases in a limited interval, and the transition appears only when there is a change in the terminal revolution number.

Finite iterative algorithms for solving generalized coupled Sylvester systems – Part I: One-sided and generalized coupled Sylvester matrix equations over generalized reflexive solutions

The generalized coupled Sylvester systems play a fundamental role in wide applications in several areas, such as stability theory, control theory, perturbation analysis, and some other fields of pure and applied mathematics. The iterative method is an important way to solve the generalized coupled Sylvester systems. In this two-part article, finite iterative methods are proposed for solving one-sided (or two-sided) and generalized coupled Sylvester matrix equations and the corresponding optimal approximation problem over generalized reflexive solutions (or reflexive solutions). In part I, an iterative algorithm is constructed to solve one-sided and coupled Sylvester matrix equations ( AY − ZB, CY − ZD) = ( E, F) over generalized reflexive matrices Y and Z. When the matrix equations are consistent, for any initial generalized reflexive matrix pair [ Y 1, Z 1], the generalized reflexive solutions can be obtained by the iterative algorithm within finite iterative steps in the absence of round-off errors, and the least Frobenius norm generalized reflexive solution pair can be obtained by choosing a special kind of initial matrix pair. The unique optimal approximation generalized reflexive solution pair [ Y ^ , Z ^ ] to a given matrix pair [ Y 0, Z 0] in Frobenius norm can be derived by finding the least-norm generalized reflexive solution pair [ Y ∼ ∗ , Z ∼ ∗ ] of two new corresponding generalized coupled Sylvester matrix equations ( A Y ∼ - Z ∼ B , C Y ∼ - Z ∼ D ) = ( E ∼ , F ∼ ) , where E ∼ = E - AY 0 + Z 0 B , F ∼ = F - CY 0 + Z 0 D . Several numerical examples are given to show the effectiveness of the presented iterative algorithm.

Finite iterative algorithms for solving generalized coupled Sylvester systems-Part II: Two-sided and generalized coupled Sylvester matrix equations over reflexive solutions

In Part I of this article, we proposed a finite iterative algorithm for the one-sided and generalized coupled Sylvester matrix equations ( AY − ZB, CY − ZD) = ( E, F) and its optimal approximation problem over generalized reflexive matrices solutions. In Part II, an iterative algorithm is constructed to solve the two-sided and generalized coupled Sylvester matrix equations ( AXB − CYD, EXF − GYH) = ( M, N), which include Sylvester and Lyapunov matrix equations as special cases, over reflexive matrices X and Y. When the matrix equations are consistent, for any initial reflexive matrix pair [ X 1, Y 1], the reflexive solutions can be obtained by the iterative algorithm within finite iterative steps in the absence of round-off errors, and the least Frobenius norm reflexive solutions can be obtained by choosing a special kind of initial matrix pair. The unique optimal approximation reflexive solution pair [ X ^ , Y ^ ] to a given matrix pair [ X 0, Y 0] in Frobenius norm can be derived by finding the least-norm reflexive solution pair [ X ∼ ∗ , Y ∼ ∗ ] of a new corresponding generalized coupled Sylvester matrix equations ( A X ∼ B - C Y ∼ D , E X ∼ F - G Y ∼ H ) = ( M ∼ , N ∼ ) , where M ∼ = M - AX 0 B + CY 0 D , N ∼ = N - EX 0 F + GY 0 H . Several numerical examples are given to show the effectiveness of the presented iterative algorithm.

The constrained inverse eigenvalue problem and its approximation for normal matrices

In this paper, the constrained inverse eigenvalue problem and associated approximation problem for normal matrices are considered. The solvability conditions and general solutions of the constrained inverse eigenvalue problem are presented, and the expression of the solution for the optimal approximation problem is obtained.

Optimum Linear Design of Two-Hop MIMO Relay Networks With QoS Requirements

This paper investigates the optimal design issue of general two-hop amplify-and-forward (AF) MIMO relay networks with one source, one destination and multiple relays, each having multiple antennas. Beginning with the study of the mean-square error (MSE) matrix of the system and the use of Wiener filter for the destination processing, a constrained optimization problem with respect to the unknown relay precoder matrix is formulated with a goal of minimizing the total relay transmit power subject to the MSE-based quality of service (QoS) requirement on each data stream. As the original optimization problem is very difficult to solve, a lower bound of the cost function is pursued to achieve an approximate optimization problem. The modified problem is then solved under the perfect channel state information (CSI) condition using the diagonalization method along with the majorization theory to obtain optimal relay precoder with either nonoptimal or optimal source precoding. It is shown that a convex relay precoding solution is available in the case of nonoptimal source precoding whereas a joint optimal source and relay precoder can be obtained through a multilevel waterfilling algorithm. The optimal design of the relay network under imperfect CSI condition is also considered by modifying the optimization problem with the channel mean and covariance matrix, leading to a robust optimization design of the relay precoder. The proposed technique is validated by Monto Carlo simulations and is also compared with one existing method.

Open mushrooms: stickiness revisited

We investigate mushroom billiards, a class of dynamical systems with sharply divided phase space. For typical values of the control parameter of the system ρ, an infinite number of marginally unstable periodic orbits (MUPOs) exist making the system sticky in the sense that unstable orbits approach regular regions in phase space and thus exhibit quasi-regular behaviour for long periods of time. The problem of finding these MUPOs is expressed as the well-known problem of finding optimal rational approximations of a real number, subject to some system-specific constraints. By introducing a generalized mushroom and using properties of continued fractions, we describe for the first time a zero measure set of control parameter values ρ ∊ (0, 1) for which all MUPOs are destroyed and therefore the system is less sticky, leading to a different power law exponent for the Poincaré recurrence time distribution statistics. The open mushroom (billiard with a hole) is then considered in order to quantify the stickiness exhibited due to MUPOs and exact leading order expressions for the algebraic decay of the survival probability function are calculated for mushrooms with triangular and rectangular stems. Numerical simulations are also performed which confirm our predictions for both sticky and less sticky mushrooms.

Kriging-based convex subspace single linkage method with path-based clustering technique for approximation-based global optimization

This paper proposes an improved approach of the Kriging-based Convex Subspace Single Linkage Method (KCSSL method), which was reported as one of approximation-based global optimization methods. The KCSSL method consists of a convex subspace clustering procedure and a local optimization procedure. For the clustering procedure, previously, the cell-based clustering technique was employed. However, this approach will involve a huge number of convexity estimations in case of a higher dimensional problem. This will cause a very high computational cost, therefore, a path-based clustering procedure is newly developed. At first, a procedure for the convexity estimation with the Kriging method is introduced. Next, outline and detailed procedure of the proposed path-based clustering technique are explained. Also, the proposed method is applied to solving some approximate optimization problems. From the numerical results, validity and effectiveness of the proposed method are discussed.

Inverse eigenvalue problems for bisymmetric matrices under a central principal submatrix constraint

This article considers an inverse eigenvalue problem for bisymmetric matrices under a central principal submatrix constraint and the corresponding optimal approximation problem. We first discuss the specified structure of bisymmetric matrices and their central principal submatrices. Then we study a special form for the matrix of independent eigenvectors for a bisymmetric matrix. Based on these, we give some necessary and sufficient conditions for the solvability of the inverse eigenvalue problem, and we derive an expression for its general solution. Finally, we obtain an expression for the solution to the corresponding optimal approximation problem.

Symmetric inverse generalized eigenvalue problem with submatrix constraints in structural dynamic model updating

In this literature, the symmetric inverse generalized eigenvalue problem with subma- trix constraints and its corresponding optimal approximation problem are studied. A necessary and sufficient condition for solvability is derived, and when solvable, the general solutions are presented.

Second order H2 optimal approximation of linear dynamical systems

AbstractIn this paper we consider the H2 optimal model reduction problem which has important applications in system approximation and has received considerable attention in the literature. An important link between special cases of this problem and rational interpolation using Krylov subspace projection methods has been recently established. We use this link to derive a solution of the second order optimal H2 approximation problem that involves the computation of all simultaneous solutions to two bivariate polynomials. We show that this is equivalent to the simultaneous solution of two 2-dimensional eigenvalue problems. To illustrate our procedure we give a few numerical examples

The solvability conditions of matrix equations with k-involution

Let m × m complex matrix P and n × n complex matrix Q be k-involutions, i.e., Pk−1 = P, Qk−1 = Q for some integer k ≥ 2. An m × n complex matrix A is (P, Q, β)symmetric if PAQ = λβA, or (P, Q, α, β)-symmetric if PAQ−α = λβA, where λ = e2πi/k and α, β ∈ {1, 2, . . . , k}. In this paper, for given matrices X, Y, E, F with appropriate sizes, the solvability of matrix equations AX = E and Y A = F under (P, Q, β)and (P, Q, α, β)-constraints, respectively, are investigated. Meanwhile, the associated optimal approximation problem is also considered when the above P and Q are unitary.

Scenario Tree Generation and Multi-Asset Financial Optimization Problems

We compare two popular scenario tree generation methods in the context of financial optimization: Moment matching and scenario reduction. Using a simple problem with a known analytic solution, we find that moment matching, when accompanied by a check to ensure the absence of arbitrage opportunities, exactly replicates this solution. On the other hand, even if the scenario trees generated by scenario reduction are arbitrage-free, the solutions to the approximate optimization problem represented by the reduced tree are highly variable. As a conclusion we strongly favor moment matching over scenario reduction for multi-asset financial optimization problems.The published version differs from this working paper version.

Sensitivity analysis of hyperbolic optimal control problems

The aim of this paper is to perform sensitivity analysis of optimal control problems defined for the wave equation. The small parameter describes the size of an imperfection in the form of a small hole or cavity in the geometrical domain of integration. The initial state equation in the singularly perturbed domain is replaced by the equation in a smooth domain. The imperfection is replaced by its approximation defined by a suitable Steklov’s type differential operator. For approximate optimal control problems the well-posedness is shown. One term asymptotics of optimal control are derived and justified for the approximate model. The key role in the arguments is played by the so called “hidden regularity” of boundary traces generated by hyperbolic solutions.

Optimization problems and second order approximated optimization problems

In this paper, a so-called second order approximated optimization problem associated to an optimization problem is considered. The equivalence between the saddle points of the lagrangian of the second order approximated optimization problem and optimal solutions of the original optimization problem is established.