Separable least squares, variable projection, and the Gauss-Newton algorithm.
Shape optimization by means of the penalty method with extrapolation
A model shape optimal design in is solved by means of the penalty method with extrapolation, which enables to obtain high order approximations of both the state function and the boundary flux, thus offering a reliable gradient for the sensitivity analysis. Convergence of the proposed method is proved for certain subsequences of approximate solutions.
Shape optimization of an elasto-perfectly plastic body
Within the range of Prandtl-Reuss model of elasto-plasticity the following optimal design problem is solved. Given body forces and surface tractions, a part of the boundary, where the (two-dimensional) body is fixed, is to be found, so as to minimize an integral of the squared yield function. The state problem is formulated in terms of stresses by means of a time-dependent variational inequality. For approximate solutions piecewise linear approximations of the unknown boundary, piecewise constant...
Shape optimization of an elasto-plastic body for the model with strain- hardening
The state problem of elasto-plasticity (for the model with strain-hardening) is formulated in terms of stresses and hardening parameters by means of a time-dependent variational inequality. The optimal design problem is to find the shape of a part of the boundary such that a given cost functional is minimized. For the approximate solutions piecewise linear approximations of the unknown boundary, piecewise constant triangular elements for the stress and the hardening parameter, and backward differences...
Shape optimization of elasto-plastic axisymmetric bodies
A minimization of a cost functional with respect to a part of a boundary is considered for an elasto-plastic axisymmetric body obeying Hencky's law. The principle of Haar-Kármán and piecewise linear stress approximations are used to solve the state problem. A convergence result and the existence of an optimal boundary is proved.
Shape optimization of elastoplastic bodies obeying Hencky's law
A minimization of a cost functional with respect to a part of the boundary, where the body is fixed, is considered. The criterion is defined by an integral of a yield function. The principle of Haar-Kármán and piecewise constant stress approximations are used to solve the state problem. A convergence result and the existence of an optimal boundary is proved.
Sharp upper global a posteriori error estimates for nonlinear elliptic variational problems
The paper is devoted to the problem of verification of accuracy of approximate solutions obtained in computer simulations. This problem is strongly related to a posteriori error estimates, giving computable bounds for computational errors and detecting zones in the solution domain where such errors are too large and certain mesh refinements should be performed. A mathematical model embracing nonlinear elliptic variational problems is considered in this work. Based on functional type estimates developed...
Simulated annealing algorithm for the minimum weighted perfect euclidean matching problem
Simulated annealing on uncorrelated energy landscapes.
Smoothing functions and algorithm for nonsymmetric circular cone complementarity problems
There has been much interest in studying symmetric cone complementarity problems. In this paper, we study the circular cone complementarity problem (denoted by CCCP) which is a type of nonsymmetric cone complementarity problem. We first construct two smoothing functions for the CCCP and show that they are all coercive and strong semismooth. Then we propose a smoothing algorithm to solve the CCCP. The proposed algorithm generates an infinite sequence such that the value of the merit function converges...
Smoothing Noisy Data with Spline Functions. Estimating the Correct Degree of Smoothing by the Method of Generalized Cross-Validation.
Solution for a classical problem in the calculus of variations via rationalized Haar functions
A numerical technique for solving the classical brachistochrone problem in the calculus of variations is presented. The brachistochrone problem is first formulated as a nonlinear optimal control problem. Application of this method results in the transformation of differential and integral expressions into some algebraic equations to which Newton-type methods can be applied. The method is general, and yields accurate results.
Solution of 3D contact shape optimization problems with Coulomb friction based on TFETI
The present paper deals with the numerical solution of 3D shape optimization problems in frictional contact mechanics. Mathematical modelling of the Coulomb friction problem leads to an implicit variational inequality which can be written as a fixed point problem. Furthermore, it is known that the discretized problem is uniquely solvable for small coefficients of friction. Since the considered problem is nonsmooth, we exploit the generalized Mordukhovich’s differential calculus to compute the needed...
Solution of degenerate parabolic variational inequalities with convection
Degenerate parabolic variational inequalities with convection are solved by means of a combined relaxation method and method of characteristics. The mathematical problem is motivated by Richard’s equation, modelling the unsaturated – saturated flow in porous media. By means of the relaxation method we control the degeneracy. The dominance of the convection is controlled by the method of characteristics.
Solution of degenerate parabolic variational inequalities with convection
Degenerate parabolic variational inequalities with convection are solved by means of a combined relaxation method and method of characteristics. The mathematical problem is motivated by Richard's equation, modelling the unsaturated – saturated flow in porous media. By means of the relaxation method we control the degeneracy. The dominance of the convection is controlled by the method of characteristics.
Solution of projection problems over polytopes.
Solvability and numerical algorithms for a class of variational data assimilation problems
A class of variational data assimilation problems on reconstructing the initial-value functions is considered for the models governed by quasilinear evolution equations. The optimality system is reduced to the equation for the control function. The properties of the control equation are studied and the solvability theorems are proved for linear and quasilinear data assimilation problems. The iterative algorithms for solving the problem are formulated and justified.
Solvability and numerical algorithms for a class of variational data assimilation problems
A class of variational data assimilation problems on reconstructing the initial-value functions is considered for the models governed by quasilinear evolution equations. The optimality system is reduced to the equation for the control function. The properties of the control equation are studied and the solvability theorems are proved for linear and quasilinear data assimilation problems. The iterative algorithms for solving the problem are formulated and justified.
Solving convex program via Lagrangian decomposition
We consider general convex large-scale optimization problems in finite dimensions. Under usual assumptions concerning the structure of the constraint functions, the considered problems are suitable for decomposition approaches. Lagrangian-dual problems are formulated and solved by applying a well-known cutting-plane method of level-type. The proposed method is capable to handle infinite function values. Therefore it is no longer necessary to demand the feasible set with respect to the non-dualized...
Solving the Task Assignment Problem with a Variable Neighborhood Search
In this paper a variable neighborhood search (VNS) approach for the task assignment problem (TAP) is considered. An appropriate neighborhood scheme along with a shaking operator and local search procedure are constructed specifically for this problem. The computational results are presented for the instances from the literature, and compared to optimal solutions obtained by the CPLEX solver and heuristic solutions generated by the genetic algorithm. It can be seen that the proposed VNS approach reaches...