Equality Type Constraints Research Articles

Stability of Solutions to Extremal Problems with Constraints Based on λ-Truncations

In this paper, we consider finite- and infinite-dimensional optimization problems with constraints of general type. We obtain sufficient conditions for stability of a strict solution and conditions for stability of a set of solutions with more than one point in it according to small perturbations of the problem parameters. For finite-dimensional extremal problems with equality-type constraints, we obtain stability conditions based on the construction of λ-truncations of mappings.

On the problem of calculus of variations with pure state constraints of equality type

On the problem of calculus of variations with pure state constraints of equality type

Well-posedness of the shooting algorithm for control-affine problems with a scalar state constraint

We deal with a control-affine problem with scalar control subject to bounds, a scalar state constraint and endpoint constraints of equality type. For the numerical solution of this problem, we propose a shooting algorithm and provide a sufficient condition for its local convergence. We exhibit an example that illustrates the theory.

Second order necessary conditions for endpoints-constrained optimal control problems on Riemannian manifolds

In this paper, we are concerned with optimal control problems evolved on Riemannian manifolds, where the initial and final states satisfy some inequality and equality type constraints, and the control set is a separable metric space. We obtain the second order necessary conditions of integral and quasi-pointwise forms, both of which work for Pontryagin type critical controls and involve the curvature tensor. For this purpose, we need to develop the needle variation technique (up to the second order) in the framework of Riemannian manifolds, and show that along a critical variational direction, a suitably chosen set related to the second order variation is convex.

A Variant of the Affine-Scaling Method for a Second-Order Cone Program

A linear cone program in which the cone is the direct product of second-order cones (Lorentz cones) is considered. For its solution, we propose a primal affine-scaling type method generalizing the corresponding method used in linear programming. The method can be considered as a special way to solve a system of necessary and sufficient optimality conditions for a pair of mutually dual cone programs. These conditions are used to derive the dependence of the dual variables on the primal variables, and the dependence is substituted into the complementarity condition. The obtained system of equations is solved with respect to the primal variables by the fixed-point iteration method. The starting points in the method belong to the cone but do not necessarily satisfy the linear equality-type constraints. The local linear convergence of the method is proved under the assumption that the solutions of the primal and dual problems are nondegenerate and strictly complementary.

Two-phase simplex method for linear semidefinite optimization

The linear semidefinite programming problem is considered. To solve it, the variant of the primal simplex method, that generalizes the corresponding method for linear programming problems, is proposed. The passage from one extreme point of the feasible set to another one is described. The main attention is given to pivoting in the case, when the extreme point is irregular, i.e. the “triangular” number of rank of the matrix in the basic point is less than number of equality type constraints in the problem. The approach for finding a starting extreme point is proposed too. The local convergence of the method is proven.

Stability analysis of the zero solution of a relay system of ordinary differential equations with two relays

We consider a relay system of ordinary differential equations whose right-hand sides are sums of linear functions and two discontinuous functions. We analyze the stability of the zero solution of a relay system of this form for the case in which the system parameters satisfy some equality-type constraints.

A parameter identification method for natural gas supply systems under unsteady gas flow

We consider the identification problem for the technical parameters, i.e., hydraulic efficiency coefficients, of gas transmission systems (GTS) under standard unsteady gas flow. Coefficient estimation is reduced to a conditional optimization problem with equality type constraints. We propose an algorithm to solve it. The algorithm's efficiency has been tested with computational experiments, including a looped gas supply system. The resulting model operates under the current level of GTS information support.

Topology optimization for elastic base under rectangular plate subjected to moving load

AbstractDistribution optimization of elastic material under elastic isotropic rectangular thin plate subjected to concentrated moving load is investigated in the present paper. The aim of optimization is to damp its vibrations in finite (fixed) time. Accepting Kirchhoff hypothesis with respect to the plate and Winkler hypothesis with respect to the base, the mathematical model of the problem is constructed as two-dimensional bilinear equation, i.e. linear in state and control function. The maximal quantity of the base material is taken as optimality criterion to be minimized. The Fourier distributional transform and the Bubnov-Galerkin procedures are used to reduce the problem to integral equality type constraints. The explicit solution in terms of two- dimensional Heaviside‘s function is obtained, describing piecewise-continuous distribution of the material. The determination of the switching points is reduced to a problem of nonlinear programming. Data from numerical analysis are presented.

Structural Optimization of an Inhomogeneous Infinite Layer in Problems on Propagation of Periodic Waves

A structural optimization problem for an infinite elastic layer inhomogeneous across its finite thickness is considered in order to find the optimum distribution functions for its elastic modulus and density that allows the propagation of a plane periodic wave with a given phase speed. The problem is mathematically formulated as a coupled system of bilinear partial differential equations of the second order with variable controlled coefficients, and the maximum absolute value of the unknown function is taken as the functional to be minimized. The optimum distribution of the elastic modulus across the layer thickness is found as a piecewise constant function. The problem is reduced to a problem of nonlinear programming under constraints of equality type. The results of numerical calculations are presented.

Aspects of hot Galilean field theory

We reconsider general aspects of Galilean-invariant thermal field theory. Using the proposal of our companion paper, we recast non-relativistic hydrodynamics in a manifestly covariant way and couple it to a background spacetime. We examine the concomitant consequences for the thermal partition functions of Galilean theories on a time-independent, but weakly curved background. We work out both the hydrodynamics and partition functions in detail for the example of parity-violating normal fluids in two dimensions to first order in the gradient expansion, finding results that differ from those previously reported in the literature. As for relativistic field theories, the equality-type constraints imposed by the existence of an entropy current appear to be in one-to-one correspondence with those arising from the existence of a hydrostatic partition function. Along the way, we obtain a number of useful results about non-relativistic hydrodynamics, including a manifestly boost-invariant presentation thereof, simplified Ward identities, the systematics of redefinitions of the fluid variables, and the positivity of entropy production.

Non-degenerate necessary optimality conditions for the optimal control problem with equality-type state constraints

In this work, an optimal control problem with state constraints of equality type is considered. Novelty of the problem formulation is justified. Under various regularity assumptions imposed on the optimal trajectory, a non-degenerate Pontryagin Maximum Principle is proven. As a consequence of the maximum principle, the Euler---Lagrange and Legendre conditions for a variational problem with equality and inequality state constraints are obtained. As an application, the equation of the geodesic curve for a complex domain is derived. In control theory, the Maximum Principle suggests the global maximum condition, also known as the Weierstrass---Pontryagin maximum condition, due to which the optimal control function, at each instant of time, turns out to be a solution to a global finite-dimensional optimization problem.

On optimal boundary and distributed control of partial integro–differential equations

Abstract A method of optimal control problems investigation for linear partial integro-differential equations of convolution type is proposed, when control process is carried out by boundary functions and right hand side of equation. Using Fourier real generalized integral transform control problem solution is reduced to minimization procedure of chosen optimality criterion under constraints of equality type on desired control function. Optimality of control impacts is obtained for two criteria, evaluating their linear momentum and total energy. Necessary and sufficient conditions of control problem solvability are obtained for both criteria. Numerical calculations are done and control functions are plotted for both cases of control process realization.

Least squares twin support vector hypersphere (LS-TSVH) for pattern recognition

The twin support vector hypersphere (TSVH) is a novel efficient pattern recognition tool, because it determines a pair of hyperspheres by solving two related SVM-type problems, each of which is smaller than in a classical SVM. In this paper we formulate a least squares version for this classifier, termed as the least squares twin support vector hypersphere (LS-TSVH). This formulation leads to extremely simple and fast algorithm for generating binary classifier based on a pair of hyperspheres. Due to equality type constraints in the formulation, the solution follows from solving two sets of nonlinear equations, instead of the two dual quadratic programming problems (QPPs) for TSVH. We show that the two sets of nonlinear equations are solved using the well-known Newton downhill algorithm. The effectiveness of proposed LS-TSVH is demonstrated by experimental results on several artificial and benchmark datasets.

Discriminatively regularized least-squares classification

Over the past decades, regularization theory is widely applied in various areas of machine learning to derive a large family of novel algorithms. Traditionally, regularization focuses on smoothing only, and does not fully utilize the underlying discriminative knowledge which is vital for classification. In this paper, we propose a novel regularization algorithm in the least-squares sense, called discriminatively regularized least-squares classification (DRLSC) method, which is specifically designed for classification. Inspired by several new geometrically motivated methods, DRLSC directly embeds the discriminative information as well as the local geometry of the samples into the regularization term so that it can explore as much underlying knowledge inside the samples as possible and aim to maximize the margins between the samples of different classes in each local area. Furthermore, by embedding equality type constraints in the formulation, the solutions of DRLSC can follow from solving a set of linear equations and the framework naturally contains multi-class problems. Experiments on both toy and real world problems demonstrate that DRLSC is often superior in classification performance to the classical regularization algorithms, including regularization networks, support vector machines and some of the recent studied manifold regularization techniques.

Optimality conditions for discrete optimal control problems

Discrete optimal control problems with variable endpoints and with equality type constraints on the control are considered. We derive first- and second-order necessary optimality conditions, which are meaningful under the assumptions weaker than those previously used in the literature.

Optimality Conditions in Discrete Optimal Control Problems with State Constraints

We consider nonlinear discrete optimal control problems with variable end points and with inequality and equality type constraints on trajectories and control. We derive first- and second-order necessary optimality conditions that are meaningful without a priori normality assumptions.

Optimality conditions for discrete calculus of variations problems

Nonlinear discrete calculus of variations problems with variable endpoints and with equality type constraints on trajectories are considered. We derive new nontrivial first- and second-order necessary optimality conditions.

Inverse function theorem and conditions of extremum for abnormal problems with non-closed range

The following two classical problems are considered: the existence and the estimate of a solution of an equation defined by a map in the neighbourhood of a point ; necessary conditions for an extremum at of a smooth function under equality-type constraints defined in terms of a non-linear map . If the range of the first derivative of at is not closed, then one cannot use classical methods of analysis based on inverse function theorems and Lagrange's principle. The results on these problems obtained in this paper are of interest in the case when the range of the first derivative of at is non-closed; these are a further development of classical results extending them to abnormal problems with non-closed range.

Augmented Lagrangian method for large-scale linear programming problems

The augmented Lagrangian and Newton methods are used to simultaneously solve the primal and dual linear programming problems. The proposed approach is applied to the primal linear programming problem with a very large number (≈106) of nonnegative variables and a moderate (≈103) number of equality-type constraints. Computation results such as the solution of a linear programme with 10 million primal variables are presented.