Necessary Conditions For Extremum Research Articles

An in-depth analysis and derivation of extremum conditions based on gradient information

This paper explores the necessary conditions for extremum in both constrained and unconstrained problems by extracting fundamental principles of constraint conditions. We provide a precise geometric understanding of the Lagrange Multiplier, prioritizing analytical insight. Beginning with a geometric interpretation of the gradient, we leverage the expansion of functions and their images to comprehend extremum and detail the Lagrangian derivation process.We expand the base vectors of the constraint surface into those of the full space and use a transition matrix to assess the function's extremum. This demonstrates how the second derivative matrix is transformed into its full-space representation to discern extrema in optimization problems. Additionally, we introduce incremental variables to optimize the second-order derivative matrix in full space, providing a novel perspective to solve extremal necessary conditions.

Nanofluid past a continuous stretching Riga sheet by Gyarmati’s principle

The work on steady laminar forced convection flow of electro magneto-hydrodynamic nanofluid over a uniformly moving sheet has been considered with the aim of evaluating non-equilibrium governing boundary layer momentum, energy and concentration equations using a classical thermodynamic variational technique. The flow, heat and mass transfer of a continuously accelerated sheet extruded in a low electrically conducting fluid, enhanced by induced ponderomotive force and diluted suspension of nanoparticles is the novel intention of the present study. The integral forms of the Lagrangian functional are constructed by determining the dual flow fields inside and on the boundary layer. Then, the necessary conditions for extremum of integral principle are derived as simple algebraic expressions in terms of boundary layer thicknesses. The thermo-physical quantities of the research interest are determined explicitly as polynomial expressions. This study examines how the skin friction coefficient, local heat and mass transfer are affected by electromagnetic force (QH), Joule heating (Ec), thermophoresis (NT) and Brownian motion (NB). The computed results indicate that electromagnetic force increases the velocity, and reduces the temperature. The concentration is increased by thermophoresis effect and decreased by Brownian diffusion. To validate the efficiency of solution procedure and confirm accuracy, certain specific results are compared with the solutions by other numerical methods available in the literature. The precision is confirmed.

Variational Rayleigh Problem of Gas Lubrication Theory. Low Compressibility Numbers

The two-dimensional variational problem for a gas-lubricated slider bearing is considered. In the gas layer the pressure field is described by the linear Reynolds equation which corresponds to low compressibility numbers. The boundary conditions are the conditions of vanishing the excess pressure on the boundaries of the domain. The load capacity acts as the functional of the variational problem. The system of necessary conditions of extremum which underlies the calculation algorithm is analyzed qualitatively. The present study develops radically and supplements the results of author’s studies at the modern level of theoretical and computational possibilities.

To the theory of variational method for Beltrami equations

We have constructed variations for classes of regular solutions of the degenerate Beltrami equation with constraints of the set-theoretic and integral types for the complex coefficient and, on this basis, proved the variational principles of maximum and other necessary conditions of extremum. Some applications to equations of mathematical physics are given.

Necessary conditions for an extremum in a mathematical programming problem

For minimization problems with equality and inequality constraints, first-and second-order necessary conditions for a local extremum are presented. These conditions apply when the constraints do not satisfy the traditional regularity assumptions. The approach is based on the concept of 2-regularity; it unites and generalizes the authors’ previous studies based on this concept.

Existence of Dual Equations by Means of Strong Necessary Conditions - Analysis of Integrability of Partial Differential Nonlinear Equations

A concept of strong necessary conditions for extremum of functional has been applied for analysis an existence of dual equations for a system of two nonlinear Partial Differential Equations (PDE) in 1+1 dimensions. We consider two types of the dual equations: the Bäcklund transformations and the Bogomolny equations. A general form of the second order PDE with a derivative-less non-linear term has been considered. In the case of a coupled system of equations the general conditions for the existence of the Bogomolny decomposition are derived. In the case of an uncoupled system of equations the Bogomolny equations become the Bäcklund transformations. It has been found a denumerable classes of coupled systems possessing the Bogomolny relationship. Weaken the method into semi-strong necessary conditions is presented together with an application to the Lax hierarchy. The method basing on both the strong and the semi-strong necessary condition concept reduces the derivation of the dual equations to an algorithm.

Necessary condition of optimality in a minimax control problem

We consider the control of bundles of trajectories of linear inclusions. Necessary conditions of extremum are derived in a form that generalizes the Pontryagin maximum principle.

Solution of a system of linear equations with uncertainty in both sides

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Necessary conditions for extremum in optimal control problems for a nonlinear elliptic equation

Necessary conditions for extremum in optimal control problems for a nonlinear elliptic equation