Finding robust yet efficient solutions to optimization problems is a major practical issue that received large attention in recent years. Starting with stochastic programming, many of the approaches to robustness lead to a significant change in the problem formulation with respect to the non-robust (nominal) case. Besides requiring a much larger computational effort, this often results into maj...

We investigate risk-averse stochastic optimization problems where riskaverse preferences are modeled with a stochastic order constraint. We propose augmented Lagrangian methods for the numerical solution of problems with multivariate and univariate stochastic order relations. The methods constructs finite-dimensional approximations of the optimization problem whose solutions converge to the sol...

Lagrangian relaxation is commonly used in combinatorial optimization to generate lower bounds for a minimization problem. We study a modified Lagrangian relaxation which generates an optimal integer solution. We call it semi-Lagrangian relaxation and illustrate its practical value by solving large-scale instances of the p-median problem.

These lecture notes review the basic properties of Lagrange multipliers and constraints in problems of optimization from the perspective of how they influence the setting up of a mathematical model and the solution technique that may be chosen. Conventional problem formulations with equality and inequality constraints are discussed first, and Lagrangian optimality conditions are presented in a ...

A large class of optimization problems can be modeled as minimization of an objective function subject to constraints given in a form of set inclusions. We discuss in this paper augmented Lagrangian duality for such optimization problems. We formulate the augmented Lagrangian dual problems and study conditions ensuring existence of the corresponding augmented Lagrange multipliers. We also discu...

A novel global optimization method based on an Augmented Lagrangian framework is introduced for continuous constrained nonlinear optimization problems. At each outer iteration k the method requires the εk-global minimization of the Augmented Lagrangian with simple constraints, where εk → ε. Global convergence to an ε-global minimizer of the original problem is proved. The subproblems are solved...

Finally the Lagranage dual function is given by g(~λ, ~ν) = inf~x L(~x,~λ, ~ν) We now make a couple of simple observations. Observation. When L(·, ~λ, ~ν) is unbounded from below then the dual takes the value −∞. Observation. g(~λ, ~ν) is concave1 as it is the infimum of a set of affine2 functions. If x is feasible solution of program (10.2)(10.4), then we have the following L(x,~λ, ~ν) = f0(x)...

In this paper, we consider the Lagrangian dual problem of a class of convex optimization problems. We first discuss the semismoothness of the Lagrangian-dual function φ. This property is then used to investigate the second-order properties of the Moreau-Yosida regularization η of the function φ, e.g., the semismoothness of the gradient g of the regularized function η. We show that φ and g are p...

The analytical target cascading (ATC) optimization technique for hierarchical systems demonstrates convergence properties only under assumptions of convexity and continuity. Many practical engineering design problems, however, involve a combination of continuous and discrete variables resulting in the development of mixed integer nonlinear programming (MINLP) formulations. While ATC has been ap...

In this paper, the Stability Augmentation System (SAS) is designed to improve the stability while Parallel Evolutionary Optimization based on Lagrangian I1 (PEvolian 11) is successfully applied to satisfy several constraints and to minimize the rising time. A controller to stabilize F-16 aircraft flying with a steady state around the altitude of 25,OOOft is described. The nonlinear pitching mot...