General Extremum Research Articles

Правило множителей Лагранжа для общей задачи на экстремум с бесконечным числом ограничений

Правило множителей Лагранжа для общей задачи на экстремум с бесконечным числом ограничений

Micromechanical analysis of volumetric growth in the context of open systems thermodynamics and configurational mechanics. Application to tumor growth

We adopt in this paper the physically and micromechanically motivated point of view that growth (resp. resorption) occurs as the expansion (resp. contraction) of initially small tissue elements distributed within a host surrounding matrix, due to the interfacial motion of their boundary. The interface motion is controlled by the availability of nutrients and mechanical driving forces resulting from the internal stresses that built in during the growth. A general extremum principle of the zero potential for open systems witnessing a change of their mass due to the diffusion of nutrients is constructed, considering the framework of open systems thermodynamics. We postulate that the shape of the tissue element evolves in such a way as to minimize the zero potential among all possible admissible shapes of the growing tissue elements. The resulting driving force for the motion of the interface sets a surface growth models at the scale of the growing tissue elements, and is conjugated to a driving force identified as the interfacial jump of the normal component of an energy momentum tensor, in line with Hadamard’s structure theorem. The balance laws associated with volumetric growth at the mesoscopic level result as the averaging of surface growth mechanisms occurring at the microscopic scale of the growing tissue elements. The average kinematics has been formulated in terms of the effective growth velocity gradient and elastic rate of deformation tensor, both functions of time. This formalism is exemplified by the simulation of the avascular growth of multicell spheroids in the presence of diffusion of nutrients, showing the respective influence of mechanical and chemical driving forces in relation to generation of internal stresses.

Differentiation of operators and optimality conditions in category interpretation

The general extremum theory essentially uses properties of operator derivatives. As an example we consider a system described by a nonlinear elliptic equation. In this system with large values of the nonlinearity parameter and the domain dimension the control-state mapping is not Gâteaux differentiable. For this reason one cannot immediately differentiate the optimality criterion and establish the necessary optimality conditions by classical methods. However the mentioned mapping is extendedly differentiable. This allows one to obtain optimality conditions imposing no constraints on system parameters. Concluding the paper, we interpret the optimality conditions with classical and extended derivatives within the theory of categories.

Estimation of Nonlinear Models with Measurement Error

This paper presents a solution to an important econometric problem, namely the root n consistent estimation of nonlinear models with measurement errors in the explanatory variables, when one repeated observation of each mismeasured regressor is available. While a root n consistent estimator has been derived for polynomial specifications (see Hausman, Ichimura, Newey, and Powell (1991)), such an estimator for general nonlinear specifications has so far not been available. Using the additional information provided by the repeated observation, the suggested estimator separates the measurement error from the “true” value of the regressors thanks to a useful property of the Fourier transform: The Fourier transform converts the integral equations that relate the distribution of the unobserved “true” variables to the observed variables measured with error into algebraic equations. The solution to these equations yields enough information to identify arbitrary moments of the “true,” unobserved variables. The value of these moments can then be used to construct any estimator that can be written in terms of moments, including traditional linear and nonlinear least squares estimators, or general extremum estimators. The proposed estimator is shown to admit a representation in terms of an influence function, thus establishing its root n consistency and asymptotic normality. Monte Carlo evidence and an application to Engel curve estimation illustrate the usefulness of this new approach.

EQUIVALENCE OF THE HIGHER ORDER ASYMPTOTIC EFFICIENCY OF k-STEP AND EXTREMUM STATISTICS

It is well known that a one-step scoring estimator that starts from any N1/2-consistent estimator has the same first-order asymptotic efficiency as the maximum likelihood estimator. This paper extends this result to k-step estimators and test statistics for k ≥ 1, higher order asymptotic efficiency, and general extremum estimators and test statistics.The paper shows that a k-step estimator has the same higher order asymptotic efficiency, to any given order, as the extremum estimator toward which it is stepping, provided (i) k is sufficiently large, (ii) some smoothness and moment conditions hold, and (iii) a condition on the initial estimator holds.For example, for the Newton–Raphson k-step estimator based on an initial estimator in a wide class, we obtain asymptotic equivalence to integer order s provided 2k ≥ s + 1. Thus, for k = 1, 2, and 3, one obtains asymptotic equivalence to first, third, and seventh orders, respectively. This means that the maximum differences between the probabilities that the (N1/2-normalized) k-step and extremum estimators lie in any convex set are o(1), o(N−3/2), and o(N−3), respectively.

Fuzzy programming and imprecise data

Abstract The parameters of mathematical programming models for many civil engineering problems can only be stated imprecisely and this leads to the formulation of fuzzy programs. Considerable progress has been made recently in the solution of such programs. The fundamental problem in a Bayesian decision analysis is the evaluation of least-biased estimates of the prior probabilities. If the prior statistical knowledge is stated crisply then the problem reduces to one of nonlinear programming. Where the prior knowledge is itself imprecise, the problem becomes one of fuzzy nonlinear programming. This leads to a general extremum principle which contains the Jaynes-Shannon formalism as a special case.

Elastic-plastic boundary element analysis as a linear complementarity problem

Elastic-plastic constitutive laws allowing for nonassociation, corners and hardening (or softening) are formulated both in the usual terms of rates and in terms of piecewise-linear approximation (local or global). The direct boundary element method combined with such descriptions of nonlinear material behaviour, is shown to reduce elasto-plastic structural analysis, both by infinitesimal and finite steps, to a linear complementary problem. On this basis, a general extremum characterization of the solution and conditions for solution uniqueness are pointed out. Various solution procedures are envisaged in the new unitary framework.

A general extremum principle for optimization problems

THE FAMOUS Pontryagin [l] maximum principle has initiated a new trend in the optimization theory. Shortly after Dubovitskij and Milyutin [2] made an attempt to give a unified general approach to optimization problems by using an intersection property of convex cones. Later Halkin and Neustadt [3] proved a very general maximum principle. Recently Gahler [4] generalized the Dubovitskij-Milyutin principle. In doing so he confronted considerable restrictions concerning the involved topological vector spaces. However, the application of convex cones is of limited use as far as practical optimization problems are concerned. The main difficulty seems to be caused by the presence of equality constraints. In this respect, Lusternik’s [S] theorem on existence of tangent directions in Banach space is rather a major step in the right direction and also permits further generalizations. The Halkin-Neustadt principle seems to be limited in scope because it depends on Brouwer’s fixed point theorem. This paper presents a very general extremum principle which is closely related to [6]. The main feature of this principle is that it has two forms (see [6]). The primary form is rather constructive and can be used to find approximate solutions of the general optimization problem. In fact, in some special cases it reduces to the well-known method of feasible directions. The dual form leads to the Euler-Lagrange equation which is well known in classical cases. Another advantage of the principle is that it includes and extends all the most important necessary conditions for optimization problems.

A support principle in a general extremum problem

A support principle in a general extremum problem

Path integral theory of anharmonic crystals

The path integral method of quantum mechanics introduced by Feynman is used to derive the recently developed selfconsistent theory of anharmonic lattice dynamics. The method allows a simple and rigorous derivation of the theory to all orders. The simplicity is obtained from integral representation of the partition function while the rigour is obtained from use of a general extremum principle rather than reliance on a variational principle. The first and second orders are derived explicitly including a derivation of the phonon frequencies via the method in each order. Close comparisons are made with the previous derivations by Choquard, Horner and Werthamer and the Green function expansion, which underline the differences in method and result.