PALHom method for QP and an application to LP. In this paper, a proximal augmented Lagrangian homotopy (PALHom) method for solving convex quadratic programming problems is proposed. This method takes the proximal augmented Lagrangian method as the outer iteration. To solve the proximal augmented Lagrangian subproblems, a homotopy method is presented as the inner iteration. The homotopy method tracks the piecewiselinear solution path of a parametric quadratic programming problem whose start problem takes an approximate solution as its solution and the target problem is the subproblem to be solved. To improve the performance of the homotopy method, the accelerated proximal gradient method is used to obtain a fairly good approximate solution that implies a good prediction of the optimal active set. Moreover, a sorting technique for the Cholesky factor update as well as an varepsilonrelaxation technique for checking primaldual feasibility and correcting the active sets are presented to improve the efficiency and robustness of the homotopy method. Simultaneously, a proximalpointbased ALHom method which is shown to converge in finite number of steps, is applied to linear programming. Numerical experiments on randomly generated problems and the problems from the CUTEr and Netlib test collections, support vector machines (SVMs) and contact problems of elasticity demonstrate that PALHom is faster than the activeset methods and the parametric active set methods and is competitive to the interiorpoint methods and the specialized algorithms designed for specific models (e.g., sequential minimal optimization (SMO) method for SVMs).

ALTRO: A Fast Solver for Constrained Trajectory Optimization Trajectory optimization is a widely used tool for robot motion planning and control. Existing solvers for these problems either rely on offtheshelf nonlinear programming solvers that are numerically robust and capable of handling arbitrary constraints, but tend to be slow because they are general purpose; or they use custom numerical methods that take advantage of the problem structure to be fast, but often lack robustness and have limited or no ability to reason about constraints. This paper presents ALTRO (Augmented Lagrangian TRajectory optimizer), a solver for constrained trajectory optimization problems that handles general nonlinear state and input constraints and offers fast convergence and numerical robustness thanks to careful exploitation of problem structure. We demonstrate its performance on a set of benchmark motionplanning problems and offer comparisons to the standard direct collocation method with largescale sequential quadratic programming and interiorpoint solvers.