On a Modified Subgradient Algorithm for Dual Problems via Sharp Augmented Lagrangian*

We study convergence properties of a modified subgradient algorithm, applied to the dual problem defined by the sharp augmented Lagrangian. The primal problem we consider is nonconvex and nondifferentiable, with equality constraints. We obtain primal and dual convergence results, as well as a condition for existence of a dual solution. Using a practical selection of the step-size parameters, we demonstrate the algorithm and its advantages on test problems, including an integer programming and an optimal control problem. *Partially Supported by 2003 UniSA ITEE Small Research Grant Ero2. Supported by CAPES, Brazil, Grant No. 0664-02/2, during her visit to the School of Mathematics and Statistics, UniSA.     

Augmented Lagrangian Duality and Nondifferentiable Optimization Methods in Nonconvex Programming

In this paper we present augmented Lagrangians for nonconvex minimization problems with equality constraints. We construct a dual problem with respect to the presented here Lagrangian, give the saddle point optimality conditions and obtain strong duality results. We use these results and modify the subgradient and cutting plane methods for solving the dual problem constructed. Algorithms proposed in this paper have some advantages. We do not use any convexity and differentiability conditions, and show that the dual problem is always concave regardless of properties the primal problem satisfies. The subgradient of the dual function along which its value increases is calculated without solving any additional problem. In contrast with the penalty or multiplier methods, for improving the value of the dual function, one need not to take the penalty like parameter to infinity in the new methods. In both methods the value of the dual function strongly increases at each iteration. In the contrast, by using the primal-dual gap, the proposed algorithms possess a natural stopping criteria. The convergence theorem for the subgradient method is also presented.     

Non-linear eigenvalue problems for <Emphasis Type="Italic">p</Emphasis>-Laplacian with variable domain

In this work we consider some eigenvalue problems for p-Laplacian with variable domain. Eigenvalues of this operator are taken as a functional of the domain. We calculate the first variation of this functional, using the obtained formula investigate behavior of the eigenvalues when the domain varies. Then we consider one shape optimization problem for the first eigenvalue, prove the necessary condition of optimality relatively domain, offer an algorithm for the numerical solution of this problem.     

Removal of numerical instability in the solution of an inverse heat conduction problem

In this paper, we consider an inverse heat conduction problem (IHCP). A set of temperature measurements at a single sensor location inside the heat conduction body is required. Using a transformation, the ill-posed IHCP becomes a Cauchy problem. Since the solution of Cauchy problem, exists and is unique but not always stable, the ill-posed problem is closely approximated by a well-posed problem. For this new well-posed problem, the existence, uniqueness, and stability of the solution are proved.     

Sulfobridged Polynuclear Complexes of Platinum(II) and Palladium(II)

When the platinum(II) and palladium(II) salts interact with ligands such as cystamine-(mercamine) HSCH₂CH₂NH₂ and 2-mercaptoethanol HSCH₂CH₂OH under certain conditions, polynuclear complexes of the compositions are obtained: [Pt₆(SCH₂CH₂NH₂)₈]Cl₄. 5H₂O and [Pd₆(SCH₂CH₂OH)₈]Cl₄. In a comparative study of the IR and X-ray spectra of synthesized complexes and ligands, as well as the results of X-ray diffraction studies, it was established that sulfur atoms of 2-mercaptoethanol occupy a bridge position with a mixed coordination of ligands in the palladium complex. In the platinum(II) complex bidentate coordination of ligands is realized through sulfur and nitrogen atoms.     

Scalarization and Nonlinear Scalar Duality for Vector Optimization with Preferences that are not necessarily a Pre-order Relation

We consider problems of vector optimization with preferences that are not necessarily a pre-order relation. We introduce the class of functions which can serve for a scalarization of these problems and consider a scalar duality based on recently developed methods for non-linear penalization scalar problems with a single constraint.     

Solution for bending of a polygonal plate with cracks by the linear conjugate method

Archive of Applied Mechanics - Stress–strain state of a polygonal multiply connected plate under bending is considered. The plate under the action of different force factors (bending moment M...     

Positive Solutions of Nonuniformly Elliptic Equations with Weighted Convex-Concave Nonlinearity

Mathematical Notes - The paper contains the proof of the existence of two different positive solutions of the problem $$\frac{\partial}{\partial z_i}\biggl(a_{ij}(z) \frac{\partial u}{\partial...     

A multi-objective programming approach to 1.5-dimensional assortment problem

In this paper we study a 1.5-dimensional cutting stock and assortment problem which includes determination of the number of different widths of roll stocks to be maintained as inventory and determination of how these roll stocks should be cut by choosing the optimal cutting pattern combinations. We propose a new multi-objective mixed integer linear programming (MILP) model in the form of simultaneously minimization two contradicting objectives related to the trim loss cost and the combined inventory cost in order to fulfill a given set of cutting orders. An equivalent nonlinear version and a particular case related to the situation when a producer is interested in choosing only a few number of types among all possible roll sizes, have also been considered. A new method called the conic scalarization is proposed for scalarizing non-convex multi-objective problems and several experimental tests are reported in order to demonstrate the validity of the developed modeling and solving approaches.