352 research outputs found
A Semismooth Predictor Corrector Method for Real-Time Constrained Parametric Optimization with Applications in Model Predictive Control
Real-time optimization problems are ubiquitous in control and estimation, and
are typically parameterized by incoming measurement data and/or operator
commands. This paper proposes solving parameterized constrained nonlinear
programs using a semismooth predictor-corrector (SSPC) method. Nonlinear
complementarity functions are used to reformulate the first order necessary
conditions of the optimization problem into a parameterized non-smooth
root-finding problem. Starting from an approximate solution, a semismooth
Euler-Newton algorithm is proposed for tracking the trajectory of the
primal-dual solution as the parameter varies over time. Active set changes are
naturally handled by the SSPC method, which only requires the solution of
linear systems of equations. The paper establishes conditions under which the
solution trajectories of the root-finding problem are well behaved and provides
sufficient conditions for ensuring boundedness of the tracking error. Numerical
case studies featuring the application of the SSPC method to nonlinear model
predictive control are reported and demonstrate the advantages of the proposed
method
A Parametric Non-Convex Decomposition Algorithm for Real-Time and Distributed NMPC
A novel decomposition scheme to solve parametric non-convex programs as they
arise in Nonlinear Model Predictive Control (NMPC) is presented. It consists of
a fixed number of alternating proximal gradient steps and a dual update per
time step. Hence, the proposed approach is attractive in a real-time
distributed context. Assuming that the Nonlinear Program (NLP) is
semi-algebraic and that its critical points are strongly regular, contraction
of the sequence of primal-dual iterates is proven, implying stability of the
sub-optimality error, under some mild assumptions. Moreover, it is shown that
the performance of the optimality-tracking scheme can be enhanced via a
continuation technique. The efficacy of the proposed decomposition method is
demonstrated by solving a centralised NMPC problem to control a DC motor and a
distributed NMPC program for collaborative tracking of unicycles, both within a
real-time framework. Furthermore, an analysis of the sub-optimality error as a
function of the sampling period is proposed given a fixed computational power.Comment: 16 pages, 9 figure
Complexity iteration analysis for strongly convex multi-objective optimization using a Newton path-following procedure
In this note, we consider the iteration complexity of solving strongly convex multi-objective optimization problems. We discuss the precise meaning of this problem, noting that its definition is ambiguous, and focus on the most natural notion of finding a set of Pareto optimal points across a grid of scalarized problems. We prove that, in most cases, performing sensitivity based path-following after obtaining one solution is the optimal strategy for this task in terms of iteration complexity
A Feedback-Based Regularized Primal-Dual Gradient Method for Time-Varying Nonconvex Optimization
This paper considers time-varying nonconvex optimization problems, utilized to model optimal operational trajectories of systems governed by possibly nonlinear physical or logical models. Algorithms for tracking a Karush-Kuhn-Tucker point are synthesized, based on a regularized primal-dual gradient method. In particular, the paper proposes a feedback-based primal-dual gradient algorithm, where analytical models for system state or constraints are replaced with actual measurements. When cost and constraint functions are twice continuously differentiable, conditions for the proposed algorithms to have bounded tracking error are derived, and a discussion of their practical implications is provided. Illustrative numerical simulations are presented for an application in power systems
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