This grant deals with research in numerical solution of nonlinear programming problems (optimization problems where an objective function of several variables is minimized in a set defined by a number of equality and inequality constraints). The objective and constraint functions are assumed smooth, nonlinear, and not necessarily convex. The goal is to find a stationary or critical point (preferably a local minimizer). It is assumed that the optimization problem is posed in finite dimension, although it might result from the discretization of an infinite dimensional optimization problem. We are not concerned with "global optimization", where the best local minimizer is sought.

Many nonlinear optimization problems, typically those arising from applications, pose several difficulties to nonlinear optimization algorithms and codes, that can include numerical instability, convergence to spurious minimizers, and slow convergence. The algorithms developed and analyzed in this project consider, depending on the origin of the problem, some of following pitfalls:

• large scale;
• high nonlinearity and non-availability of derivatives;
• mesh dependence of the optimization algorithms when applied to discretized problems;
• degeneracy (linear dependence of gradients of the active or binding constraints) and lack of strict complementarity (primal and dual complementarity variables both vanishing at the same time).

Research that is already underway includes:

• Interior-point methods, a technique used to handle the inequality constraints. The minimizer is approached through the "interior" of the feasible set. Interior-point methods are based on the application of Newton's method to a perturbed version of the first-order optimality conditions. The perturbation incorporates the (numerically efficient) notion of centrality, forcing the iterates to stay as much as possible away from the boundary of the feasible set.

  Filter methods, a scheme to enforce "global convergence" from arbitrary starting points. No "merit functions" are used to balance function value and constraint residuals. Instead, the concept of non-dominance is borrowed from multi-criteria optimization to build a "filter" able to reject poor trial iterates. The filter mechanism has been used for the first time to globalize interior-point methods:

  M. Ulbrich, S. Ulbrich, and L. N. Vicente,
  A Globally Convergent Primal-Dual Interior-Point Filter Method for Nonlinear Programming,
  Mathematical Programming, 100 (2004) 379-410. PDF
  
  
  The new primal-dual interior-point filter is being numerically tested and further studied.

• Augmented Lagrangian penalty methods, a more classical type of methods where the original problem is replaced by a sequence of penalized problems, is being studied and a new scheme for updating multipliers (dual variables) has been developed for distributed optimal control or design problems with bounds on both the control and the state variables:
  L. N. Vicente,
  Local Analysis of a New Multipliers Method,
  European Journal of Operational Research, 143 (2002) 432-451. PDF
  
  
  The new scheme considers an implicit update of the multipliers corresponding to equality constraints, allowing the possibility of using problem-dependent adjoint formulas.
  
  In the new scheme, a sequence of penalized problems is solved where only the bound constraints on the variables are left unpenalized. The solution of each penalized problem determines in turn an explicit update of the multipliers corresponding to the inequality constraints, automatically maintaining its nonnegativity.
  
  The new multipliers scheme has been also studied in the context of general NLP:
  C. P. Avelino and L. N. Vicente,
  Updating the Multipliers Associated with Inequality Constraints in an Augmented Lagrangian Multiplier Method,
  Journal of Optimization Theory and Applications, 119 (2003) 215-233. PDF