Optimal problem for the $L_{p}$ mixed chord integral

In this paper, the optimal problem for mixed $p$-capacities is investigated. The Orlicz and $L_q$ geominimal $p$-capacities are proposed and their properties, such as invariance under orthogonal matrices, isoperimetric type inequalities and cyclic type inequalities are provided as well. Moreover, the existence of the $p$-capacitary Orlicz–Petty bodies for multiple convex bodies is established, and the Orlicz and $L_q$ mixed geominimal $p$-capacities for multiple convex bodies are introduced. The continuity of the Orlicz mixed geominimal $p$-capacities and some isoperimetric type inequalities of the $L_q$ mixed geominimal $p$-capacities are proved.

In this paper, the extreme problem for Orlicz and Lq torsional rigidity is discussed. Moreover, we introduce Orlicz and Lq geominimal torsional rigidity, which is defined as being motivated through Orlicz L' mixed torsional rigidity. Also, the invariance of Orlicz and Lq geominimal torsional rigidity under orthogonal matrices is proved, and isoperimetric type inequality and circular type inequality for the torsional rigidity are established as well.

This thesis deals with the polar Orlicz-Minkowski problems and the p-capacitary Orlicz-Petty bodies. The polar Orlicz-Minkowski problems are introduced and the solvability of such problems is discussed under different conditions. In particular, under certain condition on φ; the existence of a solution is proved for a nonzero finite measure μ on Sn-1 which is not concentrated on any hemisphere of Sn-1: The existence of the p-capacitary Orlicz-Petty bodies is also established. The Orlicz and Lq geominimal capacities with respect to K0 and S0 are proposed and their properties, such as invariance under orthogonal matrices, isoperimetric type inequalities and cyclic type inequalities are provided as well.

Graph matching is a fundamental problem that arises frequently in the areas of distributed control, computer vision, and facility allocation. In this paper, we consider the optimal graph matching problem for weighted graphs, which is computationally challenging due the combinatorial nature of the set of permutations. Contrary to optimization-based relaxations to this problem, in this paper we develop a novel relaxation by constructing dynamical systems on the manifold of orthogonal matrices. In particular, since permutation matrices are orthogonal matrices with nonnegative elements, we define two gradient flows in the space of orthogonal matrices. The first minimizes the cost of weighted graph matching over orthogonal matrices, whereas the second minimizes the distance of an orthogonal matrix from the finite set of all permutations. The combination of the two dynamical systems converges to a permutation matrix which, provides a suboptimal solution to the weighted graph matching problem. Finally, our approach is shown to be promising by illustrating it on nontrivial problems.

A dynamical systems approach to weighted graph matching

Left and right inverse eigenpairs problem is a special inverse eigenvalue problem. There are many meaningful results about this problem. However, few authors have considered the left and right inverse eigenpairs problem with a submatrix constraint. In this article, we will consider the left and right inverse eigenpairs problem with the leading principal submatrix constraint for the generalized centrosymmetric matrix and its optimal approximation problem. Combining the special properties of left and right eigenpairs and the generalized singular value decomposition, we derive the solvability conditions of the problem and its general solutions. With the invariance of the Frobenius norm under orthogonal transformations, we obtain the unique solution of optimal approximation problem. We present an algorithm and numerical experiment to give the optimal approximation solution. Our results extend and unify many results for left and right inverse eigenpairs problem and the inverse eigenvalue problem of centrosymmetric matrices with a submatrix constraint.

In this paper, we discuss least squares symmetrizable solutions of matrix equations (AX = B, XC = D) and its optimal approximation solution. With the matrix row stacking, Kronecker product and special relations between two linear subspaces are topological isomorphism, and we derive the general solutions of least squares problem. With the invariance of the Frobenius norm under orthogonal transformations, we obtain the unique solution of optimal approximation problem. In addition, we present an algorithm and numerical experiment to obtain the optimal approximation solution.

In this paper, a new method for the computation of the infimum for a large class of continuous‐time H∞ optimal control problem by state feedback is presented. The main ingredients of the new method include three generalized eigenvalue problems whose coefficient matrices are from a condensed form of the given system. This condensed form is computed using only orthogonal transformations which can be implemented via a numerically stable way. The superiority of the new method over the existing one given in Chen (H∞ Control and its Applications, Chapter 5. Springer: Berlin, 1997) is verified by some numerical examples. Copyright © 2004 John Wiley & Sons, Ltd.

In this paper, the generalized orthogonal solutions to the matrix inverse problem $AX=B$ and associated optimal approximation problem are considered. The properties and structure of generalized orthogonal matrices are given, the relationships between the generalized orthogonal matrices and the orthogonal matrices are discussed. Necessary and sufficient conditions that the matrix inverse problem $AX=B$ is solvable, the general expression of solution, and it's procrustes problem are discussed. Moreover, the corresponding optimal approximation solutions are given. Finally, we give the algorithms and corresponding computational examples.

Column optimal strongly threefold orthogonal matrices in a class index eight

Algorithms for symmetric groups of simplexes

Euler wavelets method for optimal control problems of fractional integro-differential equations

Resonant geometric phases for soliton equations by M. S. Alber and J. E. Marsden Schur flows for orthogonal Hessenberg matrices by G. S. Ammar and W. B. Gragg Sub-Riemannian optimal control problems by A. M. Bloch, P. E. Crouch, and T. S. Ratiu Systems of hydrodynamic type, connected with the toda lattice and the Volterra model by O. I. Bogoyavlenskii The double bracket equation as the solution of a variational problem by R. W. Brockett Integration and visualization of matrix orbits on the connection machine by J.-P. Brunet A list of matrix flows with applications by M. T.-C. Chu The Gibbs variational principle, gradient flows, and interior-point methods by L. E. Faybusovich Optimization techniques on Riemannian manifolds by S. T. Smith On the number of real roots of a sparse polynomial system by B. Sturmfels Gradient flows for local minima of combinatorial optimization problems by W. S. Wong.

The paper presents a method for taking into account phase constraints and control constraints in the direct construction of a quasi-optimal polynomial trajectory in the state space of a system of a special form, the left-hand side of which preserves the polynomiality of the input arguments, and the right-hand side contains an orthogonal linear control transformation parameterized by the trajectory, which allows one to exclude control from tasks and build a trajectory directly in the state space. To take into account the phase constraints, we use an asymptotically exact monotone estimate of the range of values of the polynomial based on the expansion in Bernstein polynomials. Boundary conditions are taken into account using the Hermite polynomial. The presented method can also be applied to complex systems provided that they are approximated. The presented method is illustrated by the example of the task of terminal control of an aircraft.

Parallel Algorithms for LQ Optimal Control of Discrete-Time Periodic Linear Systems