Coefficient inverse extremum problems for stationary heat and mass transfer equations

A technique is developed for analyzing coefficient inverse extremum problems for a stationary model of heat and mass transfer. The model consists of the Navier-Stokes equations and the convection-diffusion equations for temperature and the pollutant concentration that are nonlinearly related via buoyancy in the Boussinesq approximation and via convective heat and mass transfer. The inverse problems are stated as the minimization of certain cost functionals at weak solutions to the original boundary value problem. Their solvability is proved, and optimality systems describing the necessary optimality conditions are derived. An analysis of the latter is used to establish sufficient conditions ensuring the local uniqueness and stability of solutions to the inverse extremum problems for particular cost functionals.     

Inverse DEA under inter-temporal dependence using multiple-objective programming

This paper deals with the inverse Data Envelopment Analysis (DEA) under inter-temporal dependence assumption. Both problems, input-estimation and output-estimation, are investigated. Necessary and sufficient conditions for input/output estimation are established utilizing Pareto and weak Pareto solutions of linear multiple-objective programming problems. Furthermore, in this paper we introduce a new optimality notion for multiple-objective programming problems, periodic weak Pareto optimality. These solutions are used in inverse DEA, and it is shown that these can be characterized by a simple modification in weighted sum scalarization tool.     

A separation theorem for nonlinear inverse images of convex sets

This paper offers first- and higher-order necessary conditions for the local disjointness of a finite system of sets that are nonlinear inverse images of convex sets. The proof is based on the characterizations of α-admissible and α-tangent variations to nonlinear inverse images of convex sets and a necessary condition for the local disjointness in terms of these variations. As an application, the results are used to obtain first- and higher-order necessary conditions of optimality in constrained optimization problems.     

Solvability of Inverse Extremal Problems for Stationary Heat and Mass Transfer Equations

We consider inverse extremal problems for the stationary system of heat and mass transfer equations describing the propagation of a substance in a viscous incompressible heat conducting fluid in a bounded domain with Lipschitz boundary. The problems consist in finding some unknown parameters of a medium or source densities from a certain information of a solution. We study solvability of the direct boundary value problem and the inverse extremal problem, justify application of the Lagrange principle, introduce and analyze the optimality systems, and establish sufficient conditions for uniqueness of solutions.     

Stability estimates in identification problems for the convection-diffusion-reaction equation

Identification problems for the stationary convection-diffusion-reaction equation in a bounded domain with a Dirichlet condition imposed on the boundary of the domain are studied. By applying an optimization method, these problems are reduced to inverse extremum problems in which the variable diffusivity and the volume density of substance sources are used as control functions. Their solvability is proved for an arbitrary weakly lower semicontinuous cost functional and particular cost functionals. An analysis of the optimality system is used to establish sufficient conditions on the input data under which the solutions of particular extremum problems are unique and stable with respect to small perturbations in the cost functional and in one of the functions involved in the boundary value problem.     

Necessary Optimality Conditions and New Optimization Methods for Cubic Polynomial Optimization Problems with Mixed Variables

Multivariate cubic polynomial optimization problems, as a special case of the general polynomial optimization, have a lot of practical applications in real world. In this paper, some necessary local optimality conditions and some necessary global optimality conditions for cubic polynomial optimization problems with mixed variables are established. Then some local optimization methods, including weakly local optimization methods for general problems with mixed variables and strongly local optimization methods for cubic polynomial optimization problems with mixed variables, are proposed by exploiting these necessary local optimality conditions and necessary global optimality conditions. A global optimization method is proposed for cubic polynomial optimization problems by combining these local optimization methods together with some auxiliary functions. Some numerical examples are also given to illustrate that these approaches are very efficient.     

半定规划的限制逆问题与广义逆问题

本文提出了半定规划的限制逆问题与广义逆问题，利用半定规划的最优性条件，分别给出了其在l∞，l1,l2模意义下的数学模型，它们仍为半定规划问题。     

Analysis and optimization in problems of cloaking of material bodies for the Maxwell equations

We consider control problems for the 3D Maxwell equations describing electromagnetic wave scattering in an unbounded inhomogeneous medium that contains a permeable isotropic obstacle with cloaking boundary. Such problems arise when studying cloaking problems by the optimization method. The boundary coefficient occurring in the impedance boundary condition plays the role of a control. We study the solvability of the control problem and derive optimality systems that describe necessary conditions for the extremum. By analyzing the constructed optimality systems, we justify sufficient conditions imposed on the input data providing the uniqueness and stability of optimal solutions.     

Sensitivity of solutions to systems of optimality conditions under the violation of constraint qualifications

A survey is given of old and new results on the sensitivity of solutions to systems of optimality conditions with respect to parametric perturbations. Results of this kind play a key role in subtle convergence analysis of various constrained optimization algorithms. General systems of optimality conditions for problems with abstract constraints, Karush-Kuhn-Tucker systems for mathematical programs, and Lagrange systems for problems with equality constraints are examined. Special attention is given to the cases where the traditional constraint qualifications are violated.     

Bounds tightening based on optimality conditions for nonconvex box-constrained optimization

First-order optimality conditions have been extensively studied for the development of algorithms for identifying locally optimal solutions. In this work, we propose two novel methods that directly exploit these conditions to expedite the solution of box-constrained global optimization problems. These methods carry out domain reduction by application of bounds tightening methods on optimality conditions. This scheme is implicit and avoids explicit generation of optimality conditions through symbolic differentation, which can be memory and time intensive. The proposed bounds tightening methods are implemented in the global solver BARON. Computational results on a test library of 327 problems demonstrate the value of our proposed approach in reducing the computational time and number of nodes required to solve these problems to global optimality.     

On stability of solutions of the coefficient inverse extremal problems for the stationary convection-diffusion equation

The coefficient inverse extremal problems are studied for the stationary convectiondiffusion equation in a bounded domain under mixed boundary conditions on the boundary of the domain. The role of control is played by the velocity vector of a medium and the functions that are involved in the boundary conditions for temperature. The solvability of the extremal problems is proven both for an arbitrary weakly lower semicontinuous quality functional and for the particular quality functionals. On the basis of analysis of the optimality system some sufficient conditions are established on the initial data providing the uniqueness and stability of optimal solutions under sufficiently small perturbations of both the quality functional and one of the functions involved in the original boundary value problem.     

A Smooth Regularization of the Projection Formula for Constrained Parabolic Optimal Control Problems

We present a smooth, that is, differentiable regularization of the projection formula that occurs in constrained parabolic optimal control problems. We summarize the optimality conditions in function spaces for unconstrained and control-constrained problems subject to a class of parabolic partial differential equations. The optimality conditions are then given by coupled systems of parabolic PDEs. For constrained problems, a non-smooth projection operator occurs in the optimality conditions. For this projection operator, we present in detail a regularization method based on smoothed sign, minimum and maximum functions. For all three cases, that is, (1) the unconstrained problem, (2) the constrained problem including the projection, and (3) the regularized projection, we verify that the optimality conditions can be equivalently expressed by an elliptic boundary value problem in the space-time domain. For this problem and all three cases we discuss existence and uniqueness issues. Motivated by this elliptic problem, we use a simultaneous space-time discretization for numerical tests. Here, we show how a standard finite element software environment allows to solve the problem and, thus, to verify the applicability of this approach without much implementation effort. We present numerical results for an example problem.     

On optimal control problems with general boundary conditions

It is shown that the necessary optimality conditions for optimal control problems with terminal constraints and with given initial state allow also to obtain in a straightforward way the necessary optimality conditions for problems involving parameters and general (mixed) boundary conditions. In a similar manner, the corresponding numerical algorithms can be adapted to handle this class of optimal control problems.This research was supported in part by the Commission on International Relations, National Academy of Sciences, under Exchange Visitor Program No. P-1-4174.The author is indebted to the anonymous reviewer bringing to his attention Ref. 9 and making him aware of the possible use of generalized inverse notation when formulating the optimality conditions.     

Duality for constrained multifacility location problems with mixed norms and applications

A dual problem is developed for the constrained multifacility minisum location problems involving mixed norms. General optimality conditions are also obtained providing new algorithms based on the concept of partial inverse of a multifunction. These algorithms which are decomposition methods, generate sequences globally converging to a primal and a dual solution respectively. Numerical results are reported.     

Optimality conditions with state constraints for semilinear systems determined by operator valued measures

In this paper we develop the necessary conditions of optimality for a class of distributed parameter systems (partial differential equations) determined by operator valued measures and controlled by vector measures. Based on some recent results on existence of optimal controls from the space of vector measures, we develop necessary conditions of optimality for a class of control problems. The main results are the necessary conditions of optimality for problems without state constraints and those with state constraints. Also, a conceptual algorithm along with a brief discussion of its convergence is presented.     

Unified approaches for solvable and unsolvable linear complementarity problems

In this paper, general linear complementarity problems (LCPs) are studied via global optimization problems. In particular, unsolvable LCPs are reformulated as multicriteria optimization, minimax optimization and quadratic programming problems. The solvability and unsolvability of LCPs are obtained via these reformulations. Furthermore, first-order and second-order global optimality conditions of LCPs are derived. Some examples are also given to demonstrate these optimality conditions.     

带跳的倒向重随机系统的最大值原理及其应用

本文研究带跳的倒向重随机系统的随机控制问题的最优性条件。在控制域为凸且控制变量进入所有系数条件下,分别以局部形式和全局形式给出必要性最优条件和充分性最优条件。把上述最大值原理应用于重随机线性二次最优控制问题,得到唯一的最优控制,并且给出应用的例子。     

A structure-preserving pivotal method for affine variational inequalities

Mathematical Programming - Affine variational inequalities (AVI) are an important problem class that subsumes systems of linear equations, linear complementarity problems and optimality conditions...     

Analysis of a Two-Dimensional Thermal Cloaking Problem on the Basis of Optimization

For a two-dimensional model of thermal scattering, inverse problems arising in the development of tools for cloaking material bodies on the basis of a mixed thermal cloaking strategy are considered. By applying the optimization approach, these problems are reduced to optimization ones in which the role of controls is played by variable parameters of the medium occupying the cloaking shell and by the heat flux through a boundary segment of the basic domain. The solvability of the direct and optimization problems is proved, and an optimality system is derived. Based on its analysis, sufficient conditions on the input data are established that ensure the uniqueness and stability of optimal solutions.