Efstratios N. Pistikopoulos - Multi-parametric Optimization and Control

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Multi-parametric Optimization and Control: краткое содержание, описание и аннотация

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R
ecent developments in multi-parametric optimization and control
Multi-Parametric Optimization and Control Researchers and practitioners can use the book as reference. It is also suitable as a primary or a supplementary textbook. Each chapter looks at the theories related to a topic along with a relevant case study. Topic complexity increases gradually as readers progress through the chapters. The first part of the book presents an overview of the state-of-the-art multi-parametric optimization theory and algorithms in multi-parametric programming. The second examines the connection between multi-parametric programming and model-predictive control—from the linear quadratic regulator over hybrid systems to periodic systems and robust control. 
The third part of the book addresses multi-parametric optimization in process systems engineering. A step-by-step procedure is introduced for embedding the programming within the system engineering, which leads the reader into the topic of the PAROC framework and software platform. PAROC is an integrated framework and platform for the optimization and advanced model-based control of process systems. 
Uses case studies to illustrate real-world applications for a better understanding of the concepts presented Covers the fundamentals of optimization and model predictive control Provides information on key topics, such as the basic sensitivity theorem, linear programming, quadratic programming, mixed-integer linear programming, optimal control of continuous systems, and multi-parametric optimal control An appendix summarizes the history of multi-parametric optimization algorithms. It also covers the use of the parametric optimization toolbox (POP), which is comprehensive software for efficiently solving multi-parametric programming problems.

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Let картинка 90be the global minimum of problem ( 1.12), and that the gradient of the equality constraints are linearly independent. In addition, assume that the corresponding Lagrange multiplier is Multiparametric Optimization and Control - изображение 91. The vector Multiparametric Optimization and Control - изображение 92is a perturbation vector. The solution of problem ( 1.12) is a function of the perturbation vector along with the multiplier. Hence, the Lagrange function can be written as

(1.13) Calculating the partial derivative of the Lagrange function with respect to the - фото 93

Calculating the partial derivative of the Lagrange function with respect to the perturbation vector, we have

(1.14) Multiparametric Optimization and Control - изображение 94

which yields

(1.15) Multiparametric Optimization and Control - изображение 95

Hence, the Lagrange multipliers can be interpreted as a measure of sensitivity of the objective function with respect to the perturbation vector of the constraints at the optimum point картинка 96.

1.2 Concepts of Multi‐parametric Programming

1.2.1 Basic Sensitivity Theorem

Having the essentials of optimization for the purposes of this book covered, the objective of this subchapter is to introduce the role of parameters in an optimization formulation. In this context, the following multi‐parametric programming problem is considered:

(1.16) where is the vector of the continuous optimization variables is the v - фото 97

where картинка 98is the vector of the continuous optimization variables, картинка 99is the vector of the uncertain parameters, and the sets картинка 100, картинка 101correspond to the inequality and equality constraint sets, respectively.

Theorem 1.1 (Basic Sensitivity Theorem, [1])

Let a general multi‐parametric programming problem be described by (1.16). Assume that the functions defining problem (1.16) are twice differentiable in and their gradients with respect to and the constraints are once continuously differentiable in in a neighborhood of . In addition, assume that the second‐order sufficient conditions for a local minimum of the problem hold at with associated Lagrange multipliers and . Lastly, let the gradients (for such that ) and be linearly independent (i.e. LICQ holds), and for such that , i.e. strict complementary slackness (SCS) holds.

Then, the first‐order sensitivity results for a second‐order local minimizing point are known as the basic sensitivity theorem (BST), and the following properties hold:

is a local isolated minimizing point of the problem and the associated Lagrange multipliers and are unique.

For in the neighborhood or , there exists a unique, but continuously differentiable vector function satisfying the second‐order sufficient conditions for a local minimum of the problem with associated unique Lagrange multipliers and .

For near the set of binding inequalities is unchanged, SCS holds and the binding constraint gradients are linearly independent at .

Proof

See [1].

If there exist Lagrange multipliers, картинка 102and such that the firstorder KKT conditions hold then we have 117 and the - фото 103, such that the first‐order KKT conditions hold, then we have:

(1.17) Multiparametric Optimization and Control - изображение 104

and the vector Multiparametric Optimization and Control - изображение 105is defined as follows:

(1.18) Furthermore if there exists for which 119 the B - фото 106

Furthermore, if there exists for which 119 the Basic Sensitivity Theorem holds and it is identically - фото 107for which

(1.19) the Basic Sensitivity Theorem holds and it is identically satisfied for a - фото 108

the Basic Sensitivity Theorem holds, and it is identically satisfied for a neighborhood картинка 109around картинка 110and can be differentiated with respect to to yield explicit expressions for the partial derivatives of the vector - фото 111to yield explicit expressions for the partial derivatives of the vector function The firstorder estimate of the variation of an isolated local solution x - фото 112.

The first‐order estimate of the variation of an isolated local solution x( картинка 113) of ( 1.16) and the associated unique Lagrange multipliers and can be approximated given that is known and that - фото 114and can be approximated given that is known and that is available In part - фото 115can be approximated, given that is known and that is available In particular let be the concatenation - фото 116is known and that картинка 117is available.

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