1,198 research outputs found
Sequential Convex Programming Methods for Solving Nonlinear Optimization Problems with DC constraints
This paper investigates the relation between sequential convex programming
(SCP) as, e.g., defined in [24] and DC (difference of two convex functions)
programming. We first present an SCP algorithm for solving nonlinear
optimization problems with DC constraints and prove its convergence. Then we
combine the proposed algorithm with a relaxation technique to handle
inconsistent linearizations. Numerical tests are performed to investigate the
behaviour of the class of algorithms.Comment: 18 pages, 1 figur
Some Applications of Polynomial Optimization in Operations Research and Real-Time Decision Making
We demonstrate applications of algebraic techniques that optimize and certify
polynomial inequalities to problems of interest in the operations research and
transportation engineering communities. Three problems are considered: (i)
wireless coverage of targeted geographical regions with guaranteed signal
quality and minimum transmission power, (ii) computing real-time certificates
of collision avoidance for a simple model of an unmanned vehicle (UV)
navigating through a cluttered environment, and (iii) designing a nonlinear
hovering controller for a quadrotor UV, which has recently been used for load
transportation. On our smaller-scale applications, we apply the sum of squares
(SOS) relaxation and solve the underlying problems with semidefinite
programming. On the larger-scale or real-time applications, we use our recently
introduced "SDSOS Optimization" techniques which result in second order cone
programs. To the best of our knowledge, this is the first study of real-time
applications of sum of squares techniques in optimization and control. No
knowledge in dynamics and control is assumed from the reader
Recent Advances in Computational Methods for the Power Flow Equations
The power flow equations are at the core of most of the computations for
designing and operating electric power systems. The power flow equations are a
system of multivariate nonlinear equations which relate the power injections
and voltages in a power system. A plethora of methods have been devised to
solve these equations, starting from Newton-based methods to homotopy
continuation and other optimization-based methods. While many of these methods
often efficiently find a high-voltage, stable solution due to its large basin
of attraction, most of the methods struggle to find low-voltage solutions which
play significant role in certain stability-related computations. While we do
not claim to have exhausted the existing literature on all related methods,
this tutorial paper introduces some of the recent advances in methods for
solving power flow equations to the wider power systems community as well as
bringing attention from the computational mathematics and optimization
communities to the power systems problems. After briefly reviewing some of the
traditional computational methods used to solve the power flow equations, we
focus on three emerging methods: the numerical polynomial homotopy continuation
method, Groebner basis techniques, and moment/sum-of-squares relaxations using
semidefinite programming. In passing, we also emphasize the importance of an
upper bound on the number of solutions of the power flow equations and review
the current status of research in this direction.Comment: 13 pages, 2 figures. Submitted to the Tutorial Session at IEEE 2016
American Control Conferenc
Linearly Solvable Stochastic Control Lyapunov Functions
This paper presents a new method for synthesizing stochastic control Lyapunov
functions for a class of nonlinear stochastic control systems. The technique
relies on a transformation of the classical nonlinear Hamilton-Jacobi-Bellman
partial differential equation to a linear partial differential equation for a
class of problems with a particular constraint on the stochastic forcing. This
linear partial differential equation can then be relaxed to a linear
differential inclusion, allowing for relaxed solutions to be generated using
sum of squares programming. The resulting relaxed solutions are in fact
viscosity super/subsolutions, and by the maximum principle are pointwise upper
and lower bounds to the underlying value function, even for coarse polynomial
approximations. Furthermore, the pointwise upper bound is shown to be a
stochastic control Lyapunov function, yielding a method for generating
nonlinear controllers with pointwise bounded distance from the optimal cost
when using the optimal controller. These approximate solutions may be computed
with non-increasing error via a hierarchy of semidefinite optimization
problems. Finally, this paper develops a-priori bounds on trajectory
suboptimality when using these approximate value functions, as well as
demonstrates that these methods, and bounds, can be applied to a more general
class of nonlinear systems not obeying the constraint on stochastic forcing.
Simulated examples illustrate the methodology.Comment: Published in SIAM Journal of Control and Optimizatio
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