Constraints Of Inequality Type Research Articles

Topology optimization of extruded beams modeled with the XFEM for maximizing their natural frequencies

In this paper, an efficient topology optimization approach is developed for maximizing the fundamental natural frequency of extruded beams. Mass fraction and static compliance bounds are defined using inequality-type constraints in the optimization problem. An XFEM approach, previously proposed by the authors for analyzing beam elements, is extended herein to compute the natural frequencies of the beam. The method allows for 3D modeling of beams with a significant reduction in the number of degrees of freedom and therefore also yields efficient optimization procedure. This reduction is made possible by incorporating global enrichment functions in the longitudinal direction, which enables a significant reduction in the number of elements in that direction without loss of accuracy. A nonlinear optimization problem is formulated using continuous density-based design variables that represent the material distribution in the beam’s cross-section. The optimization problem is then solved using a gradient-based approach with analytical sensitivities. The well-known Solid Isotropic Material with Penalization (SIMP) method is used to acquire discrete solutions. We study the optimal design of short and long beams. It is shown that for short beams, localized vibration modes appear within the cross-section, leading to a significant distortion deformation mode of the cross-section. The optimized design of the long beam shows global deformation modes with an increase of 15% in the fundamental frequency compared with a non-optimized design consisting of a hollow rectangular cross-section with the same mass.

Elasticity Problem with a Cusp between Thin Inclusion and Boundary

This paper concerns an equilibrium problem for an an elastic body with a thin rigid inclusion crossing an external boundary of the body at zero angle. The inclusion is assumed to be exfoliated from the surrounding elastic material that provides an interfacial crack. To avoid nonphysical interpenetration of the opposite crack faces, we impose inequality type constraints. Moreover, boundary conditions at the crack faces depend on a positive parameter describing a cohesion. A solution existence of the problem with different conditions on the external boundary is proved. Passages to the limit are analyzed as the damage parameter tends to infinity and to zero. Finally, an optimal control problem with a suitable cost functional is investigated. In this case, a part of the rigid inclusion is located outside of the elastic body, and a control function is a shape of the inclusion.

MATHEMATICAL MODEL OF THE MARKET OF ONE PRODUCT WITH OPTIMAL DELIVERY TO THE MARKET UNDER CONDITIONS OF DELAY

MATHEMATICAL MODEL OF THE MARKET OF ONE PRODUCT WITH OPTIMAL DELIVERY TO THE MARKET UNDER CONDITIONS OF DELAY

Controllability for Problems with Mixed Constraints

For a controlled system with mixed equality- and inequality-type constraints anda geometric constraint that is a nonempty closed convex set, sufficient conditions for the existenceof feasible positional controls are obtained in terms of first derivatives of mappings that define themixed constraints. In addition, in terms of the first and second derivatives of these mappings,sufficient conditions for the existence of feasible positional controls are found that are alsoapplicable in the case of degeneration of the first derivatives of the mappings.

On the Brachistochrone problem with state constraints on the slope angle

The problem of maximizing the horizontal coordinate of a point moving in a vertical plane under the action of gravity forces, viscous friction, and the interrelated Brachistochrone problem are investigated. It is assumed that inequality-type constraints are imposed on the trajectory inclination angle. The optimal control problem with state constraints is reduced to the problem with control constraints. As a result, the sequence and the number of the reaching the phase constraints for the original problem are determined and the synthesis of the optimal control is designed.

A Sequential Quadratic Programming Method for Constrained Multi-objective Optimization Problems

In this article, a globally convergent sequential quadratic programming (SQP) method is developed for multi-objective optimization problems with inequality type constraints. A feasible descent direction is obtained using a linear approximation of all objective functions as well as constraint functions. The sub-problem at every iteration of the sequence has feasible solution. A non-differentiable penalty function is used to deal with constraint violations. A descent sequence is generated which converges to a critical point under the Mangasarian-Fromovitz constraint qualification along with some other mild assumptions. The method is compared with a selection of existing methods on a suitable set of test problems.

On a modified optimal control problem with first-order PDE constraints and the associated saddle-point optimality criterion

In this paper, based on a new multidimensional optimal control problem, in short (OCP), we introduce a modified multidimensional optimal control problem (which is simpler or easier to solve than the original one) involving first-order partial differential equations (PDEs) and inequality-type constraints. Optimality conditions associated with this new optimal control problem are formulated and proved, as well. Furthermore, under some generalized convexity assumptions, we establish an equivalence between an optimal solution of (OCP) and a saddle-point associated with the Lagrange functional corresponding to the modified multidimensional optimal control problem. Also, in order to illustrate the main results and their effectiveness, some applications are presented.

Regularized Extragradient Method of Finding a Solution to an Optimal Control Problem with Inaccurately Specified Input Data

We consider an optimal control problem described by a system of linear ordinary differential equations with boundary conditions of general form defined by inequality-type constraints in the case when the input data are inaccurately specified. In general, such problems are unstable with respect to perturbations of the input data and require the development of special stable solution methods. In this paper we propose a regularized variant of the extragradient method, study its convergence, and construct a regularizing operator.

Investigation of Controllability and Regularity Conditions for State Constrained Problems * *This publication was supported by the Russian Foundation for Basic Research, project no. 16-31-60005, and by the Ministry of Education and Science of the Russian Federation (Agreement number 02.a03.21.0008 and project no. 1.962.2017). The results of Section 2 were obtained by the first author under the support of the

Abstract A new method to prove non-degenerate optimality conditions in optimal control problems with state constraints of inequality type is proposed. This method involves the second-order derivative of the state constraint function and is based on the notion of the extended Hamilton-Pontryagin function and on the related optimality conditions.

Necessary conditions for a weak minimum in a general optimal control problem with integral equations on a variable time interval

We study an optimal control problem with a nonlinear Volterra-type integral equation considered on a nonfixed time interval, subject to endpoint constraints of equality and inequality type, mixed state-control constraints of inequality and equality type, and pure state constraints of inequality type. The main assumption is the linear–positive independence of the gradients of active mixed constraints with respect to the control. We obtain first-order necessary optimality conditions for an extended weak minimum, the notion of which is a natural generalization of the notion of weak minimum with account of variations of the time. The conditions obtained generalize the corresponding ones for problems with ordinary differential equations.

Necessary second-order conditions for a weak local minimum in a problem with endpoint and control constraints

One of the first steps towards necessary second-order optimality conditions in problems with constraints was taken by Dubovitskii and Milyutin in 1965. They offered a scheme that was very effective in smooth optimization problems, but seemed to be not suitable for applications in problems with pointwise control constraints. In this article we consider a modification of the Dubovitskii–Milyutin scheme, which allows to derive necessary second-order conditions for a weak local minimum in optimal control problems with a finite number of endpoint constraints of equality and inequality type and with pointwise control constraints of inequality type given by smooth functions. Assuming that the gradients of active control constraints are linearly independent, we provide rather straightforward proof of these conditions for a measurable and essentially bounded optimal control.

Necessary Conditions for a Weak Minimum in Optimal Control Problems with Integral Equations Subject to State and Mixed Constraints

The first order necessary optimality conditions for a weak minimum are derived for optimal control problems with Volterra-type integral equations, considered on a fixed time interval, subject to endpoint constraints of equality and inequality type, mixed state-control constraints of inequality and equality type, and pure state constraints of inequality type. The main assumption is the uniform linear-positive independence of the gradients of active mixed constraints with respect to the control. The conditions obtained generalize the Euler--Lagrange equation (as a stationarity condition) for the Lagrange problem in the classical calculus of variations with ordinary differential equations.

Задача со смещением для вырождающегося внутри области гиперболического уравнения

For a degenerate hyperbolic equation in characteristic region (lune) a boundary-value problem with operators of fractional integro-differentiation is studied. The solution of this equation on the characteristics is related point-to-point to the solution and its derivative on the degeneration line. The uniqueness theorem is proved by the modified Tricomi method with inequality-type constraints on the known functions. Question of the problem solution's existence is reduced to the solvability of a singular integral equation with Cauchy kernel of the normal type.

Minimal surfaces with constraints of inequality type

In this paper we establish sufficient conditions for bifurcations of minimal surfaces with nonstrict inequality constraints.

On a boundary value problem with shift for an equation of mixed type in an unbounded domain

We study the unique solvability of a problem with shift for an equation of mixed type in an unbounded domain. We prove the uniqueness theorem under inequality-type constraints for known functions for various orders of the fractional differentiation operators in the boundary condition. The existence of a solution is proved by reduction to a Fredholm equation of the second kind, whose unconditional solvability follows from the uniqueness of the solution of the problem.

New Accurate and Flexible Design Procedure for a Stable KWTA Continuous Time Network

The classical continuous time recurrent (Hopfield) network is considered and adapted to K -winner-take-all operation. The neurons are of sigmoidal type with a controllable gain G, an amplitude m and interconnected by the conductance p. The network is intended to process one by one a sequence of lists, each of them with N distinct elements, each of them squeezed to [0,I] admission interval, each of them having an imposed minimum separation between elements z(min). The network carries out: 1) a matching dynamic process between the order of list elements and the order of outputs, and 2) a binary type steady-state separation between K and K+1 outputs, the former surpassing a +ξ threshold and the later falling under the -ξ threshold. As a result, the machine will signal the ranks of the K largest elements of the list. To achieve 1), the initial condition of processing phase has to be placed in a computable θ -vicinity of zero-state. This requires a resetting procedure after each list. To achieve 2) the bias current M has to be within a certain interval computable from circuit parameters. In addition, the steady-state should be asymptotically stable. To these goals, we work with high gain and exploit the sigmoid properties and network symmetry. The various inequality type constraints between parameters are shown to be compatible and a neat synthesis procedure, simple and flexible, is given for the tanh sigmoid. It starts with the given parameters N, K, I, z(min), m and computes simple bounds of p, G, ξ, θ, and M. Numerical tests and comments reveal qualities and shortcomings of the method.

Second-order sufficient optimality conditions for control problems with linearly independent gradients of control constraints

Second-order sufficient conditions of a bounded strong minimum are derived for optimal control problems of ordinary differential equations with initial-final state constraints of equality and inequality type and control constraints of inequality type. The conditions are stated in terms of quadratic forms associated with certain tuples of Lagrange multipliers. Under the assumption of linear independence of gradients of active control constraints they guarantee the bounded strong quadratic growth of the so-called “violation function”. Together with corresponding necessary conditions they constitute a no-gap pair of conditions.

Generalized convexity and efficiency for non-regular multiobjective programming problems with inequality-type constraints

For multiobjective problems with inequality-type constraints the necessary conditions for efficient solutions are presented. These conditions are applied when the constraints do not necessarily satisfy any regularity assumptions, and they are based on the concept of 2-regularity introduced by Izmailov. In general, the necessary optimality conditions are not sufficient and the efficient solution set is not the same as the Karush–Kuhn–Tucker points set. So it is necessary to introduce generalized convexity notions. In the multiobjective non-regular case we give the notion of 2-KKT-pseudoinvex-II problems. This new concept of generalized convexity is both necessary and sufficient to guarantee the characterization of all efficient solutions based on the optimality conditions.

Shape optimization in contact problems of the theory of elasticity with incomplete external loading data

The contact interaction without friction of an absolutely rigid punch with an elastic half-space is considered. The external loads on the elastic medium are not fixed in advance, but a set containing all the admissible forms of applied forces is assumed to be specified. Using a guaranteed (minimax) approach, problems of optimizing the shape of the punch from the condition that its mass is a minimum are formulated. Inequality-type constraints, imposed on the total force and moments applied to the punch from the elastic-medium side, are assumed. Using Betti's reciprocal theorem and calculating the “worst” case for different types of constraints, the corresponding forces are determined and the optimum shape of the punch is obtained in analytical form.

Generalized convexity in non-regular programming problems with inequality-type constraints

Convexity plays a very important role in optimization for establishing optimality conditions. Different works have shown that the convexity property can be replaced by a weaker notion, the invexity. In particular, for problems with inequality-type constraints, Martin defined a weaker notion of invexity, the Karush–Kuhn–Tucker-invexity (hereafter KKT-invexity), that is both necessary and sufficient to obtain Karush–Kuhn–Tucker-type optimality conditions. It is well known that for this result to hold the problem has to verify a constraint qualification, i.e., it must be regular or non-degenerate. In non-regular problems, the classical optimality conditions are totally inapplicable. Meaningful results were obtained for problems with inequality-type constraints by Izmailov. They are based on the 2-regularity condition of the constraints at a feasible point. In this work, we generalize Martin's result to non-regular problems by defining an analogous concept, the 2-KKT-invexity, and using the characterization of the tangent cone in the 2-regular case and the necessary optimality condition given by Izmailov.