On the finite termination of an entropy function based smoothing Newton method for vertical linear complementarity problems

By using a smooth entropy function to approximate the non-smooth max-type function, a vertical linear complementarity problem (VLCP) can be treated as a family of parameterized smooth equations. A Newton-type method with a testing procedure is proposed to solve such a system. We show that the proposed algorithm finds an exact solution of VLCP in a finite number of iterations, under some conditions milder than those assumed in literature. Some computational results are included to illustrate the potential of this approach.Newton method;Finite termination;Entropy function;Smoothing approximation;Vertical linear complementarity problems

Homotopy interpretation of price adjustment proces

Price Theory;econometrics

On the finite termination of an entropy function based smoothing Newton method for vertical linear complementarity problems

By using a smooth entropy function to approximate the non-smooth max-type function, a vertical linear complementarity problem (VLCP) can be treated as a family of parameterized smooth equations. A Newton-type method with a testing procedure is proposed to solve such a system. We show that the proposed algorithm finds an exact solution of VLCP in a finite number of iterations, under some conditions milder than those assumed in literature. Some computational results are included to illustrate the potential of this approach

Hessian barrier algorithms for linearly constrained optimization problems

In this paper, we propose an interior-point method for linearly constrained optimization problems (possibly nonconvex). The method - which we call the Hessian barrier algorithm (HBA) - combines a forward Euler discretization of Hessian Riemannian gradient flows with an Armijo backtracking step-size policy. In this way, HBA can be seen as an alternative to mirror descent (MD), and contains as special cases the affine scaling algorithm, regularized Newton processes, and several other iterative solution methods. Our main result is that, modulo a non-degeneracy condition, the algorithm converges to the problem's set of critical points; hence, in the convex case, the algorithm converges globally to the problem's minimum set. In the case of linearly constrained quadratic programs (not necessarily convex), we also show that the method's convergence rate is $\mathcal{O}(1/k^\rho)$ for some $\rho\in(0,1]$ that depends only on the choice of kernel function (i.e., not on the problem's primitives). These theoretical results are validated by numerical experiments in standard non-convex test functions and large-scale traffic assignment problems.Comment: 27 pages, 6 figure

On the Finite Termination of An Entropy Function Based Smoothing Newton Method for Vertical Linear Complementarity Problems

By using a smooth entropy function to approximate the non-smooth max-type function, a vertical linear complementarity problem (VLCP) can be treated as a family of parameterized smooth equations. A Newton-type method with a testing procedure is proposed to solve such a system. We show that the proposed algorithm finds an exact solution of VLCP in a finite number of iterations, under some conditions milder than those assumed in literature. Some computational results are included to illustrate the potential of this approach

Societal bargaining and stability

Collective Bargaining;labour economics

Robot Motion Planning Under Topological Constraints

My thesis addresses the the problem of manipulation using multiple robots with cables. I study how robots with cables can tow objects in the plane, on the ground and on water, and how they can carry suspended payloads in the air. Specifically, I focus on planning optimal trajectories for robots. Path planning or trajectory generation for robotic systems is an active area of research in robotics. Many algorithms have been developed to generate path or trajectory for different robotic systems. One can classify planning algorithms into two broad categories. The first one is graph-search based motion planning over discretized configuration spaces. These algorithms are complete and quite efficient for finding optimal paths in cluttered 2-D and 3-D environments and are widely used [48]. The other class of algorithms are optimal control based methods. In most cases, the optimal control problem to generate optimal trajectories can be framed as a nonlinear and non convex optimization problem which is hard to solve. Recent work has attempted to overcome these shortcomings [68]. Advances in computational power and more sophisticated optimization algorithms have allowed us to solve more complex problems faster. However, our main interest is incorporating topological constraints. Topological constraints naturally arise when cables are used to wrap around objects. They are also important when robots have to move one way around the obstacles rather than the other way around. Thus I consider the optimal trajectory generation problem under topological constraints, and pursue problems that can be solved in finite-time, guaranteeing global optimal solutions. In my thesis, I first consider the problem of planning optimal trajectories around obstacles using optimal control methodologies. I then present the mathematical framework and algorithms for multi-robot topological exploration of unknown environments in which the main goal is to identify the different topological classes of paths. Finally, I address the manipulation and transportation of multiple objects with cables. Here I consider teams of two or three ground robots towing objects on the ground, two or three aerial robots carrying a suspended payload, and two boats towing a boom with applications to oil skimming and clean up. In all these problems, it is important to consider the topological constraints on the cable configurations as well as those on the paths of robot. I present solutions to the trajectory generation problem for all of these problems

On the existence of a positive definite solution of the matrix equation X+AXA=I

Matrices;mathematics