Nonlinear contraction tools for constrained optimization

Abstract

This thesis derives new results linking nonlinear contraction analysis, a recent stability theory for nonlinear systems, and constrained optimization theory. Although dynamic systems and optimization are both areas that have been extensively studied [21], few results have been achieved in this direction because strong enough tools for dynamic systems were not available. Contraction analysis provides the necessary mathematical background. Based on an operator that projects the speed of the system on the tangent space of the constraints, we derive generalizations of Lagrange parameters. After presenting some initial examples that show the relations between contraction and optimization, we derive a contraction theorem for nonlinear systems with equality constraints. The method is applied to examples in differential geometry and biological systems. A new physical interpretation of Lagrange parameters is provided. In the autonomous case, we derive a new algorithm to solve minimization problems. Next, we state a contraction theorem for nonlinear systems with inequality constraints. In the autonomous case, the algorithm solves minimization problems very fast compared to standard algorithms. Finally, we state another contraction theorem for nonlinear systems with time-varying equality constraints. A new generalization of time varying Lagrange parameters is given. In the autonomous case, we provide a solution for a new class of optimization problems, minimization with time-varying constraints.