A Primal-Dual Augmented Lagrangian

Nonlinearly constrained optimization problems can be solved by minimizing a sequence of simpler unconstrained or linearly constrained subproblems. In this paper, we discuss the formulation of subproblems in which the objective is a primal-dual generalization of the Hestenes-Powell augmented Lagrangian function. This generalization has the crucial feature that it is minimized with respect to both the primal and the dual variables simultaneously. A benefit of this approach is that the quality of the dual variables is monitored explicitly during the solution of the subproblem. Moreover, each subproblem may be regularized by imposing explicit bounds on the dual variables. Two primal-dual variants of conventional primal methods are proposed: a primal-dual bound constrained Lagrangian (pdBCL) method and a primal-dual $\ell$1 linearly constrained Lagrangian (pd$\ell$1-LCL) method

Global convergence of a stabilized sequential quadratic semidefinite programming method for nonlinear semidefinite programs without constraint qualifications

In this paper, we propose a new sequential quadratic semidefinite programming (SQSDP) method for solving nonlinear semidefinite programs (NSDPs), in which we produce iteration points by solving a sequence of stabilized quadratic semidefinite programming (QSDP) subproblems, which we derive from the minimax problem associated with the NSDP. Differently from the existing SQSDP methods, the proposed one allows us to solve those QSDP subproblems just approximately so as to ensure global convergence. One more remarkable point of the proposed method is that any constraint qualifications (CQs) are not required in the global convergence analysis. Specifically, under some assumptions without CQs, we prove the global convergence to a point satisfying any of the following: the stationary conditions for the feasibility problem; the approximate-Karush-Kuhn-Tucker (AKKT) conditions; the trace-AKKT conditions. The latter two conditions are the new optimality conditions for the NSDP presented by Andreani et al. (2018) in place of the Karush-Kuhn-Tucker conditions. Finally, we conduct some numerical experiments to examine the efficiency of the proposed method

Real-Time Motion Planning of Legged Robots: A Model Predictive Control Approach

We introduce a real-time, constrained, nonlinear Model Predictive Control for the motion planning of legged robots. The proposed approach uses a constrained optimal control algorithm known as SLQ. We improve the efficiency of this algorithm by introducing a multi-processing scheme for estimating value function in its backward pass. This pass has been often calculated as a single process. This parallel SLQ algorithm can optimize longer time horizons without proportional increase in its computation time. Thus, our MPC algorithm can generate optimized trajectories for the next few phases of the motion within only a few milliseconds. This outperforms the state of the art by at least one order of magnitude. The performance of the approach is validated on a quadruped robot for generating dynamic gaits such as trotting.Comment: 8 page