Lecture

Nonlinear Optimization Methods

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In course

ME-454: Modelling and optimization of energy systems

The goal of the lecture is to present and apply techniques for the modelling and the thermo-economic optimisation of industrial process and energy systems. The lecture covers the problem statement, the solving methods for the simulation and the single and ...

Description

This lecture introduces various methods for solving nonlinear optimization problems, starting with direct and indirect search approaches, followed by heuristic methods. The instructor explains how to transform inequality constraints into equality constraints using slack variables and discusses the Newton-Raphson method for solving nonlinear equations. The lecture also covers the Lagrange equivalence and the branch and bound method for handling integer variables in optimization problems.

Instructor

François Maréchal

Ph D. in engineering Chemical process engineer Researcher and lecturer in the field of computer aided process and energy systems engineering. Lecturer in the mechanical engineering, electrical engineering and environmental sciences engineering in EPFL. I'm responsible for the Minor in Energy of EPFL and I'm involved in 3 projects of the Competence Center in Energy and Mobility (2nd generation biofuel, Wood SOFC, and gas turbine development with CO2 mitigation) in which i'm contributing to the energy conversion system design and optimisation. Short summary of my scientific carrer After a graduation in chemical engineering from the University of Liège, I have obtained a Ph. D. from the University of Liège in the LASSC laboratory of Prof. Kalitventzeff (former president of the European working party on computer aided process engineering). This laboratory was one of the pioneering laboratory in the field of Computer Aided Process Engineering. In the group of Professor Kalitventzeff, I have worked on the development and the applications of data reconciliation, process modelling and optimisation techniques in the chemical process industry, my experience ranges from nuclear power stations to chemical plants. In the LASSC, I have been responsible from the developments in the field of rational use of energy in the industry. My first research topic has been the methodological development of process integration techniques, combining the use of pinch based methods and of mathematical programming: e.g. for the design of multiperiod heat exchanger networks or Mixed integer non linear programming techniques for the optimal management of utility systems. Fronted with applications in the industry, my work then mainly concentrated on the optimal integration of utility systems considering not only the energy requirements but the cost of the energy requirements and the energy conversion systems. I developed methods for analysing and integrating the utility system, the steam networks, combustion (including waste fuel), gas turbines or other advanced energy conversion systems (cogeneration, refrigeration and heat). The techniques applied uses operation research tools like mixed integer linear programming and exergy analysis. In order to evaluate the results of the utility integration, a new graphical method for representing the integration of the utility systems has been developed. By the use of MILP techniques, the method developed for the utility integration has been extended to handled site scale problems, to incorporate environmental constraints and reduce the water usage. This method (the Effect Modelling and Optimisation method) has been successfully applied to the chemical plants industry, the pulp and paper industry and the power plant. Instead of focusing on academic problems, I mainly developed my research based on industrial applications that lead to valuable and applicable patented results. Recently the methods developed have been extended to realise the thermoeconomic optimisation of integrated systems like fuel cells. My present R&D work concerns the application of multi-objective optimisation strategies in the design of processes and integrated energy conversion systems. Since 2001, Im working in the Industrial Energy Systems Laboratory (LENI) of Ecole Polytechnique fédérale de Lausanne (EPFL) where Im leading the R&D activities in the field of Computer Aided Analysis and Design of Industrial Energy Systems with a major focus on sustainable energy conversion system development using thermo-economic optimisation methodologies. A part from the application and the development of process integration techniques, that remains my major field of expertise, the applications concern : Rational use of water and energy in Industrial processes and industrial production sites : projects with NESTLE, EDF, VEOLIA and Borregaard (pulp and paper).Energy conversion and process design : biofuels from waste biomass (with GASNAT, EGO and PSI), water dessalination and waste water treatment plant (VEOLIA), power plant design (ALSTOM), Energy conversion from geothermal sources (BFE). Integrated energy systems in urban areas : together with SCANE and SIG (GE) and IEA annexe 42 for micro-cogeneration systems. I as well contributed to the definition of the 2000 Watt society and to studies concerning the emergence of green technologies on the market in the frame of the Alliance for Global Sustainability.

Official source

Ontological neighbourhood

Theoretical computer science

Algorithms and data structures: Combinatorial optimization

Related concepts (30)

Constrained optimization

In mathematical optimization, constrained optimization (in some contexts called constraint optimization) is the process of optimizing an objective function with respect to some variables in the presence of constraints on those variables. The objective function is either a cost function or energy function, which is to be minimized, or a reward function or utility function, which is to be maximized.

Mathematical optimization

Mathematical optimization (alternatively spelled optimisation) or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries.

Nonlinear programming

In mathematics, nonlinear programming (NLP) is the process of solving an optimization problem where some of the constraints or the objective function are nonlinear. An optimization problem is one of calculation of the extrema (maxima, minima or stationary points) of an objective function over a set of unknown real variables and conditional to the satisfaction of a system of equalities and inequalities, collectively termed constraints. It is the sub-field of mathematical optimization that deals with problems that are not linear.

Branch and bound

Branch and bound (BB, B&B, or BnB) is a method for solving optimization problems by breaking them down into smaller sub-problems and using a bounding function to eliminate sub-problems that cannot contain the optimal solution. It is an algorithm design paradigm for discrete and combinatorial optimization problems, as well as mathematical optimization. A branch-and-bound algorithm consists of a systematic enumeration of candidate solutions by means of state space search: the set of candidate solutions is thought of as forming a rooted tree with the full set at the root.

Duality (optimization)

In mathematical optimization theory, duality or the duality principle is the principle that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem. If the primal is a minimization problem then the dual is a maximization problem (and vice versa). Any feasible solution to the primal (minimization) problem is at least as large as any feasible solution to the dual (maximization) problem.

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