A note on characterizing solution set of nonsmooth pseudoinvex optimization problem

In this paper, by making use of some examples, we first point out some results are not true in the earlier works of Mishra and Lai (On Characterization of Solution Sets of Nonsmooth Pseudoinvex Minimization Problems. IEEE Computer Society, pp. 739–741, 2009). Furthermore, we correct and modify corresponding results. In the end, we give an example to illustrate the main result. Our study modifies and improves some previously known results.     

What makes an optimization problem hand?

We address the question: “Are some classes of combinatorial optimization problems intrinsically harder than others, without regard to the algorithm one uses, or can difficulty only be assessed relative to particular algorithms?” We provide a measure of the hardness of a particular optimization problem for a particular optimization algorithm and present two algorithm-independent quantities that use this measure to provide answers to our question. In the first of these we average hardness over all possible algorithms and show that according to this quantity, there are no distinctions between optimization problems. In this sense no problems are intrinsically harder than others. For the second quantity, rather than average over all algorithms, we consider the level of hardness of a problem (or class of problems) for the associated optimal algorithm. By this criteria there are classes of problems that are intrinsically harder than others.     

Explicit solution of the Carlson problem

In this paper we give new, explicit and direct proof of the classical Carlson problem, by using only elementary matrix completion tools and combinatorics.     

Explicit solution to a linear-quadratic optimal control problem with an arbitrary terminal

An optimal control problem of minimization of the sum of a linear-quadratic integral functional and a terminal of arbitrary form is considered. It is proved that in this case a control can be also constructed in an explicit analytic form.     

Explicit solution for a vector-valued LQG homing problem

In this note, a vector-valued LQG homing problem subject to a controlled stock-wealth process is formulated. Under certain conditions, the problem is solved explicitly via an auxiliary scalar LQG homing problem.     

Efficient solution of interval optimization problem

In this paper the interval valued function is defined in the parametric form and its properties are studied. A methodology is developed to study the existence of the solution of a general interval optimization problem, which is expressed in terms of the interval valued functions. The methodology is applied to the interval valued convex quadratic programming problem.     

Explicit solution of the row completion problem for matrix pencils

In this paper we give new, explicit and simplified conditions for the problem of determining the possible strict equivalence class of a pencil with prescribed rows (columns). This improves the result from [1] and solves an important particular case of the challenge problem posed by Loiseau et al. [7]. All results that we present are obtained over arbitrary fields.     

The statement and numerical solution of an optimization problem in X-ray tomography

A nonclassical problem is considered for the transport equation with coefficients depending on the energy of radiation. The task is to find the discontinuity surfaces for the coefficients of the equation from measurements of the radiation flux leaving the medium. For this tomography problem, an optimization problem is stated and numerically analyzed. The latter consists in determining the radiation energy that ensures the best reconstruction of the unknown medium. A simplified optimization problem is solved analytically.     

Explicit solution of the inverse kinematic problem in the non-Herglotz case

The inverse kinematic problem is solved in the half space R ₊ ^ν+1 ={(x,z)|z?0,x∈R^ν, ν?1 under the assumption that the index of refraction can be represented in the form $$n^2 (x,z) = K^2 (z) + \sum\limits_{j = 1}^\nu {\Phi _j^2 (x_j ),} n_z< 0.$$ . The solution obtained is a generalization of the Herglotz-Wiechert formula. A formula is presented for the solution of the inverse kinematic problem in the general case of separation of variables in the eikonal equation.     

A proper flow chart for a numerical solution of an inverse problem by an optimization method

Two flow charts are presented to solve the same inverse problem by an optimization method. Numerical examples are given to show that the first flow chart, which is frequently opted for by researchers, is computationally much more costly than the second one. This is explained by the necessity to use a dense partition net and increase the number of minimization iterations for decreasing the residual functional down to a certain value.     

On an optimization problem of Bellman

This paper considers a problem proposed by Bellman in 1970: given a continuous kernel K(x, y) defined on I × I, find a pair of continuous functions f and g such that f(x) + g(y) ? K(x, y) on I × I and ∝_I (f + g) is minimum. The notion of basic decomposition of K is defined, and it is shown that whenever K(x, y) or K(x, a + b ? y), I = [a, b], admits a basic decomposition, Bellman's problem has a unique differentiable solution, provided K is differentiable. Explicit formulas for such solutions are given. More generally, there are kernels which admit basic decompositions on subintervals which can be “pasted together” to define a unique piecewise differentiable solution.     

Explicit series solution for the Glauert-jet problem by means of the homotopy analysis method

In this article, the self-similar wall jet over an impermeable, resting plane surface (the Glauert-jet) is considered. Through an analytic technique to solve nonlinear problems namely the homotopy analysis method, we obtain an explicit series solution for the Glauert-jet problem. This series solution converges efficiently to the closed-form solution found by Glauert in the whole region 0 ⩽ ξ < +∞. In the frame of the homotopy analysis method, it is shown that the convergence region of the explicit series solution may be adjusted to obtain more accurate results.     

Polynomial hulls and an optimization problem

We say that a subset of Cⁿ is hypoconvex if its complement is the union of complex hyperplanes. We say it is strictly hypoconvex if it is smoothly bounded hypoconvex and at every point of the boundary the real Hessian of its defining function is positive definite on the complex tangent space at that point. Let B_n be the open unit ball in Cⁿ.Suppose K is a C^∞ compact manifold in ∂B₁ × Cⁿ, n > 1, diffeomorphic to ∂B₁ × ∂B_n, each of whose fibers K_z over ∂B₁ bounds a strictly hypoconvex connected open set. Let K be the polynomialhull of K. Then we show that K∖K is the union of graphs of analytic vector valued functions on B₁. This result shows that an unnatural assumption regarding the deformability of K in an earlier version of this result is unnecessary. Next, we study an H^∞ optimization problem. If pis a C^∞ real-valued function on ∂B₁× Cⁿ, we show that the infimum γ_ρ = inf_ƒ∈H ^∞ (B₁)ⁿ ‖ρ(z, ƒ (z))‖_∞ is attained by a unique bounded ƒ provided that the set (z, w) ∈ ∂B₁ × C ⁿ|ρ(z, w) ≤ γ_ρ has bounded connected strictly hypoconvex fibers over the circle.     

Saddle problem and optimization problem as an integrated system

A parametric problem of convex programming and a linear optimization problem on a convex set are considered as an integrated system of problems. Properties of this system are studied, the sphere of its application is discussed, and methods for its solution are proposed. The convergence of the proposed methods is established.     

A solution to anisotropic suboptimal filtering problem by convex optimization

This paper considers a robust filtering problem for a linear discrete time invariant system with measured and estimated outputs. The system is exposed to random disturbances with imprecisely known distributions generated by an unknown stable shaping filter from the Gaussian white noise. The stochastic uncertainty of the input disturbance is measured by the mean anisotropy functional. The estimation error is quantified by the anisotropic norm which is a stochastic analogue of the H _∞ norm. A sufficient condition for an estimator to exist and ensure that the error is less than a given threshold value is derived in form of a convex inequality on the determinant of a positive definite matrix and two linear matrix inequalities. The suboptimal problem setting results to a set of the estimators ensuring the anisotropic norm of the error to be strictly bounded thereby providing some additional degree of freedom to impose some additional constraints on the estimator performance specification.     

Parametric solution to the joint system identification and optimization problem

This study is concerned with the simultaneous identification and optimization of static systems. The necessity and the advantages of an integrated approach to the identification and optimization of the system model is established theoretically as well as computationally. A parametric approach to the integrated problem is proven to converge to the integrated problem solution. The general methodology of decomposition of large-scale systems is extended by implementingfeasible decomposition of the joint problem. A multilevel approach is then utilized to successfully solve example problems. Handling the system constraints via a penalty-function technique is shown to be an efficient approach when using the parametric formulation of the joint problem. Numerical results for two example problems are presented using the Univac 1108 digital computer, revealing the economic advantages and disadvantages of the integrated approach to the identification and optimization problems.The authors wish to thank Messieurs I. Lefkowitz, L. S. Lasdon, F. Gembicki, and O. B. Olagundoye for their critique, comments, and suggestions during the course of this study.     

Explicit solution of a Kolmogorov equation

Ever since the technique of the Kalman-Bucy filter was popularized, there has been an intense interest in finding new classes of finite-dimensional recursive filters. In the late seventies, the concept of the estimation algebra of a filtering system was introduced. It has been the major tool in studying the Duncan-Mortensen-Zakai equation. Recently the second author has constructed general finite-dimensional filters which contain both Kalman-Bucy filters and Benes filter as special cases. In this paper we consider a filtering system with arbitrary nonlinear driftf(x) which satisfies some regularity assumption at infinity. This is a natural assumption in view of Theorem 10 of [DTWY] in a special case. Under the assumption on the observation h(x)=constant, we propose writing down the solution of the Duncan-Mortensen-Zakai equation explicitly.This research was supported by Army Grant DAAH-04-93G-0006.     

Explicit solution of the operator equation

In this paper we find the explicit solution of the equation

A^*X+X^*A=B