On sequential optimality conditions for smooth constrained optimization

Sequential optimality conditions provide adequate theoretical tools to justify stopping criteria for nonlinear programming solvers. Approximate KKT and Approximate Gradient Projection conditions are analyzed in this work. These conditions are not necessarily equivalent. Implications between different conditions and counter-examples will be shown. Algorithmic consequences will be discussed. Article Download View On sequential optimality conditions for … Read more

Global minimization using an Augmented Lagrangian method with variable lower-level constraints

A novel global optimization method based on an Augmented Lagrangian framework is introduced for continuous constrained nonlinear optimization problems. At each outer iteration the method requires the $\varepsilon$-global minimization of the Augmented Lagrangian with simple constraints. Global convergence to an $\varepsilon$-global minimizer of the original problem is proved. The subproblems are solved using the $\alpha$BB … Read more

On Second-Order Optimality Conditions for Nonlinear Programming

Necessary Optimality Conditions for Nonlinear Programming are discussed in the present research. A new Second-Order condition is given, which depends on a weak constant rank constraint requirement. We show that practical and publicly available algorithms (www.ime.usp.br/~egbirgin/tango) of Augmented Lagrangian type converge, after slight modifications, to stationary points defined by the new condition. Article Download View … Read more

Continuous Optimization Methods for Structure Alignments

Structural Alignment is an important tool for fold identification of proteins, structural screening on ligand databases, pharmacophore identification and other applications. In the general case, the optimization problem of superimposing two structures is nonsmooth and nonconvex, so that most popular methods are heuristic and do not employ derivative information. Usually, these methods do not admit … Read more

Low Order-Value Optimization and Applications

Given r real functions F1 (x), . . . , Fr (x) and an integer p between 1 and r, the Low Order- Value Optimization problem (LOVO) consists of minimizing the sum of the functions that take the p smaller values. If (y1 , . . . , yr ) is a vector of data … Read more

Density-based Globally Convergent Trust-Region Methods for Self-Consistent Field Electronic Structure Calculations

A theory of globally convergent trust-region methods for self-consistent field electronic structure calculations that use the density matrices as variables is developed. The optimization is performed by means of sequential global minimizations of a quadratic model of the true energy. The global minimization of this quadratic model, subject to the idempotency of the density matrix … Read more

On Augmented Lagrangian methods with general lower-level constraints

Augmented Lagrangian methods with general lower-level constraints are considered in the present research. These methods are useful when efficient algorithms exist for solving subproblems where the constraints are only of the lower-level type. Two methods of this class are introduced and analyzed. Inexact resolution of the lower-level constrained subproblems is considered. Global convergence is proved … Read more

Nonlinear-Programming Reformulation of the Order-Value Optimization problem

Order-value optimization (OVO) is a generalization of the minimax problem motivated by decision-making problems under uncertainty and by robust estimation. New optimality conditions for this nonsmooth optimization problem are derived. An equivalent mathematical programming problem with equilibrium constraints is deduced. The relation between OVO and this nonlinear-programming reformulation is studied. Particular attention is given to … Read more

Augmented Lagrangian methods under the Constant Positive Linear Dependence constraint qualification

Two Augmented Lagrangian algorithms for solving KKT systems are introduced. The algorithms differ in the way in which penalty parameters are updated. Possibly infeasible accumulation points are characterized. It is proved that feasible limit points that satisfy the Constant Positive Linear Dependence constraint qualification are KKT solutions. Boundedness of the penalty parameters is proved under … Read more