25 research outputs found
On the piecewise-concave approximations of functions
The piecewise-concave function may be used to approximate a wide range of
other functions to arbitrary precision over a bounded set. In this short paper,
this property is proven for three function classes: (a) the multivariate twice
continuously differentiable function, (b) the univariate Lipschitz-continuous
function, and (c) the multivariate separable Lipschitz-continuous function.Comment: 4 pages; written as a supplement to submitted journal pape
Sufficient Conditions for Feasibility and Optimality of Real-Time Optimization Schemes - II. Implementation Issues
The idea of iterative process optimization based on collected output
measurements, or "real-time optimization" (RTO), has gained much prominence in
recent decades, with many RTO algorithms being proposed, researched, and
developed. While the essential goal of these schemes is to drive the process to
its true optimal conditions without violating any safety-critical, or "hard",
constraints, no generalized, unified approach for guaranteeing this behavior
exists. In this two-part paper, we propose an implementable set of conditions
that can enforce these properties for any RTO algorithm. This second part
examines the practical side of the sufficient conditions for feasibility and
optimality (SCFO) proposed in the first and focuses on how they may be enforced
in real application, where much of the knowledge required for the conceptual
SCFO is unavailable. Methods for improving convergence speed are also
considered.Comment: 56 pages, 15 figure
Sufficient Conditions for Feasibility and Optimality of Real-Time Optimization Schemes - I. Theoretical Foundations
The idea of iterative process optimization based on collected output
measurements, or "real-time optimization" (RTO), has gained much prominence in
recent decades, with many RTO algorithms being proposed, researched, and
developed. While the essential goal of these schemes is to drive the process to
its true optimal conditions without violating any safety-critical, or "hard",
constraints, no generalized, unified approach for guaranteeing this behavior
exists. In this two-part paper, we propose an implementable set of conditions
that can enforce these properties for any RTO algorithm. The first part of the
work is dedicated to the theory behind the sufficient conditions for
feasibility and optimality (SCFO), together with their basic implementation
strategy. RTO algorithms enforcing the SCFO are shown to perform as desired in
several numerical examples - allowing for feasible-side convergence to the
plant optimum where algorithms not enforcing the conditions would fail.Comment: Working paper; supplementary material available at:
http://infoscience.epfl.ch/record/18807
Implementation techniques for the SCFO experimental optimization framework
The material presented in this document is intended as a comprehensive,
implementation-oriented supplement to the experimental optimization framework
presented in a companion document. The issues of physical degradation, unknown
Lipschitz constants, measurement/estimation noise, gradient estimation,
sufficient excitation, and the handling of soft constraints and/or a numerical
cost function are all addressed, and a robust, implementable version of the
sufficient conditions for feasible-side global convergence is proposed.Comment: supplementary document; 66 page
On linear and quadratic Lipschitz bounds for twice continuously differentiable functions
Lower and upper bounds for a given function are important in many
mathematical and engineering contexts, where they often serve as a base for
both analysis and application. In this short paper, we derive piecewise linear
and quadratic bounds that are stated in terms of the Lipschitz constants of the
function and the Lipschitz constants of its partial derivatives, and serve to
bound the function's evolution over a compact set. While the results follow
from basic mathematical principles and are certainly not new, we present them
as they are, from our experience, very difficult to find explicitly either in
the literature or in most analysis textbooks.Comment: 3 pages; supplementary documen