82 research outputs found
Primal-Dual Enumeration for Multiparametric Linear Programming
Optimal control problems for constrained linear systems with a linear cost can be posed as multiparametric linear programs (pLPs) and solved explicitly offline. Several algorithms have recently been proposed in the literature that solve these pLPs in a fairly efficient manner, all of which have as a base operation the computation and removal of redundant constraints. For many problems, it is this redundancy elimination that requires the vast majority of the computation time. This paper introduces a new solution technique for multiparametric linear programs based on the primal–dual paradigm. The proposed approach reposes the problem as the vertex enumeration of a linearly transformed polytope and then simultaneously computes both its vertex and halfspace representations. Exploitation of the halfspace representation allows, for smaller problems, a very significant reduction in the number of redundancy elimination operations required, resulting in many cases in a much faster algorithm
Reduced Memory Footprint in Multiparametric Quadratic Programming by Exploiting Low Rank Structure
In multiparametric programming an optimization problem which is dependent on
a parameter vector is solved parametrically. In control, multiparametric
quadratic programming (mp-QP) problems have become increasingly important since
the optimization problem arising in Model Predictive Control (MPC) can be cast
as an mp-QP problem, which is referred to as explicit MPC. One of the main
limitations with mp-QP and explicit MPC is the amount of memory required to
store the parametric solution and the critical regions. In this paper, a method
for exploiting low rank structure in the parametric solution of an mp-QP
problem in order to reduce the required memory is introduced. The method is
based on ideas similar to what is done to exploit low rank modifications in
generic QP solvers, but is here applied to mp-QP problems to save memory. The
proposed method has been evaluated experimentally, and for some examples of
relevant problems the relative memory reduction is an order of magnitude
compared to storing the full parametric solution and critical regions
Probabilistic structural analysis by extremum methods
The objective is to demonstrate discrete extremum methods of structural analysis as a tool for structural system reliability evaluation. Specifically, linear and multiobjective linear programming models for analysis of rigid plastic frames under proportional and multiparametric loadings, respectively, are considered. Kinematic and static approaches for analysis form a primal-dual pair in each of these models and have a polyhedral format. Duality relations link extreme points and hyperplanes of these polyhedra and lead naturally to dual methods for system reliability evaluation
Lexicographic perturbation for multiparametric linear programming with applications to control
Accepted versio
An improved multi-parametric programming algorithm for flux balance analysis of metabolic networks
Flux balance analysis has proven an effective tool for analyzing metabolic
networks. In flux balance analysis, reaction rates and optimal pathways are
ascertained by solving a linear program, in which the growth rate is maximized
subject to mass-balance constraints. A variety of cell functions in response to
environmental stimuli can be quantified using flux balance analysis by
parameterizing the linear program with respect to extracellular conditions.
However, for most large, genome-scale metabolic networks of practical interest,
the resulting parametric problem has multiple and highly degenerate optimal
solutions, which are computationally challenging to handle. An improved
multi-parametric programming algorithm based on active-set methods is
introduced in this paper to overcome these computational difficulties.
Degeneracy and multiplicity are handled, respectively, by introducing
generalized inverses and auxiliary objective functions into the formulation of
the optimality conditions. These improvements are especially effective for
metabolic networks because their stoichiometry matrices are generally sparse;
thus, fast and efficient algorithms from sparse linear algebra can be leveraged
to compute generalized inverses and null-space bases. We illustrate the
application of our algorithm to flux balance analysis of metabolic networks by
studying a reduced metabolic model of Corynebacterium glutamicum and a
genome-scale model of Escherichia coli. We then demonstrate how the critical
regions resulting from these studies can be associated with optimal metabolic
modes and discuss the physical relevance of optimal pathways arising from
various auxiliary objective functions. Achieving more than five-fold
improvement in computational speed over existing multi-parametric programming
tools, the proposed algorithm proves promising in handling genome-scale
metabolic models.Comment: Accepted in J. Optim. Theory Appl. First draft was submitted on
August 4th, 201
On Degeneracy Issues in Multi-parametric Programming and Critical Region Exploration based Distributed Optimization in Smart Grid Operations
Improving renewable energy resource utilization efficiency is crucial to
reducing carbon emissions, and multi-parametric programming has provided a
systematic perspective in conducting analysis and optimization toward this goal
in smart grid operations. This paper focuses on two aspects of interest related
to multi-parametric linear/quadratic programming (mpLP/QP). First, we study
degeneracy issues of mpLP/QP. A novel approach to deal with degeneracies is
proposed to find all critical regions containing the given parameter. Our
method leverages properties of the multi-parametric linear complementary
problem, vertex searching technique, and complementary basis enumeration.
Second, an improved critical region exploration (CRE) method to solve
distributed LP/QP is proposed under a general mpLP/QP-based formulation. The
improved CRE incorporates the proposed approach to handle degeneracies. A
cutting plane update and an adaptive stepsize scheme are also integrated to
accelerate convergence under different problem settings. The computational
efficiency is verified on multi-area tie-line scheduling problems with various
testing benchmarks and initial states
Polyhedral Tools for Control
Polyhedral operations play a central role in constrained control. One of the most fundamental operations is that of projection, required both by addition and multiplication. This thesis investigates projection and its relation to multi-parametric linear optimisation for the types of problems that are of particular interest to the control community. The first part of the thesis introduces an algorithm for the projection of polytopes in halfspace form, called Equality Set Projection (ESP). ESP has the desirable property of output sensitivity for non-degenerate polytopes. That is, a linear number of linear programs are needed per output facet of the projection. It is demonstrated that ESP is particularly well suited to control problems and comparative simulations are given, which greatly favour ESP. Part two is an investigation into the multi-parametric linear program (mpLP). The mpLP has received a lot of attention in the control literature as certain model predictive control problems can be posed as mpLPs and thereby pre-solved, eliminating the need for online optimisation. The structure of the solution to the mpLP is studied and an approach is pre- sented that eliminates degeneracy. This approach causes the control input to be continuous, preventing chattering, which is a significant problem in control with a linear cost. Four new enumeration methods are presented that have benefits for various control problems and comparative simulations demonstrate that they outperform existing codes. The third part studies the relationship between projection and multi-parametric linear programs. It is shown that projections can be posed as mpLPs and mpLPs as projections, demonstrating the fundamental nature of both of these problems. The output of a multi-parametric linear program that has been solved for the MPC control inputs offline is a piecewise linear controller defined over a union of polyhedra. The online work is then to determine which region the current measured state is in and apply the appropriate linear control law. This final part introduces a new method of searching for the appropriate region by posing the problem as a nearest neighbour search. This search can be done in logarithmic time and we demonstrate speed increases from 20Hz to 20kHz for a large example system
An Efficient Algorithm for Vertex Enumeration of Arrangement
This paper presents a state-of-the-art algorithm for the vertex enumeration
problem of arrangements, which is based on the proposed new pivot rule, called
the Zero rule. The Zero rule possesses several desirable properties: i) It gets
rid of the objective function; ii) Its terminal satisfies uniqueness; iii) We
establish the if-and-only if condition between the Zero rule and its valid
reverse, which is not enjoyed by earlier rules; iv) Applying the Zero rule
recursively definitely terminates in steps, where is the dimension of
input variables. Because of so, given an arbitrary arrangement with
vertices of hyperplanes in , the algorithm's complexity is at
most and can be as low as if it is
a simple arrangement, while Moss' algorithm takes , and
Avis and Fukuda's algorithm goes into a loop or skips vertices because the
if-and-only-if condition between the rule they chose and its valid reverse is
not fulfilled. Systematic and comprehensive experiments confirm that the Zero
rule not only does not fail but also is the most efficient
Optimal Vehicle Charging in Bilevel Power-Traffic Networks via Charging Demand Function
Electric vehicle (EV) charging couples the operation of power and traffic
networks. Specifically, the power network determines the charging price at
various locations, while EVs on the traffic network optimize the charging power
given the price, acting as price-takers. We model such decision-making
processes by a bilevel program, with the power network at the upper-level and
the traffic network at the lower-level. However, since the two networks are
managed by separate entities and the charging expense term, calculated as the
product of charging price and charging demand, is nonlinear. Solving the
bilevel program is nontrivial. To overcome these challenges, we derive the
charging demand function using multiparametric programming theory. This
function establishes a piecewise linear relationship between the charging price
and the optimal charging power, enabling the power network operator to manage
EV charging power independently while accounting for the coupling between the
two networks. With the derived function, we are also able to replace the
nonlinear charging expense term with a piecewise quadratic one, thus
guaranteeing solution optimality. Our numerical studies demonstrate that
different traffic demands can have an impact on charging patterns and the power
network can effectively incentivize charging at low-price nodes through price
setting.Comment: submitted to IEEE Transactions on Smart Gri
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