Saddle Point Seeking for Convex Optimization Problems

A constrained extremum‐seeking control approach

“…Conditions C1 and C2 establish the existence of a unique minimizer for problem (6). In the section, we formulate a barrier function method that can be used to approximate the local minimizer u by the solution of an unconstrained optimization problem.…”

“…/ of the augmented unconstrained cost (7) that approximates to an O. / neighbourhood of the minimizer of the static nonlinear optimization problem (6).…”

“…An alternative proof of this theorem for unconstrained ESC can also be found in [9]. In this section, we provide an analysis of the proposed constrained extremum-control system design applied to an unknown dynamical system (1) to find an equilibrium that solves the real-time optimization problem (6). The closed-loop ESC system is given by…”

“…In this approach, a barrier function or interior-point method was used to enforce constraints and feasibility of the closed-loop trajectories. A Lagrangian, saddle-point, ESC technique is proposed in [6]. A similar approach is proposed in [7] to handle a class of stochastic control systems.…”

Shahshahani gradient-like extremum seeking

“…For instance, in Tan, Li, and Mareels (2013), the problem of extremum seeking with saturation constraints is addressed based on penalty functions and antiwindup techniques. Also, in Durr, Zeng, and Ebenbauer (2013), an extremum seeking algorithm based on a Lie-bracket approximation is shown to achieve convex optimization of static maps.…”