Overview: Optimization is the study of problems in which we wish to optimize (either
maximize or minimize) a function (usually of several variables) often subject to a
collection of restrictions on these variables.
The restrictions are known as constraints
and the function to be optimized is the objective function. Optimization problems are
widespread in the modelling of real world systems, and cover a very broad range of
applications.
Problems of engineering design (such as the design of electronic circuits
subject to a tolerancing and tuning provision), information technology (such as the
extraction of meaningful information from large databases and the classication of
data), nancial decision making and investment planning (such as the selection of
optimal investment portfolios), and transportation management and so on arise in
the form of a multi-variable optimization problem or an optimal control problem.
Introduction: What is an optimization problem? Areas of applications of optimization.
Modelling of real life optimization problems.
Multi-variable optimization. Formulation of multi-variable optimization problems; Struc-
ture of optimization problems: objective functions and constraints. Mathematical
background: multi-variable calculus and linear algebra; (strict) local and (strict)
global minimizers and maximizers; convex sets, convex and concave functions; global
extrema and uniqueness of solutions.
Optimality conditions: First and second order conditions for unconstrained prob-
lems; Lagrange multiplier conditions for equality constrained problems; Kuhn-Tucker
conditions for inequality constrained problems.
Numerical Methods for Unconstrained Problems: Steepest descent method,
Newton's method, Conjugate gradient methods.
Numerical Methods for Constrained Problems: Penalty Methods.
Optimal Control: What is an optimal control problem? Areas of applications of optimal
control. Mathematical background: ordinary differential equations and systems of
linear differential equations.
The Pontryagin maximum principle: Autonomous control problems; unbounded
controls
Optimisation is the process of finding the best possible solution to a problem or task. It is important because it helps improve efficiency, reduce costs, and achieve desired outcomes.
The key principles of optimisation include defining the problem, identifying constraints, setting goals, collecting and analyzing data, developing and testing solutions, and implementing the most effective solution.
When choosing a book for optimisation, consider the author's credentials and expertise, the level of detail and examples provided, and the relevance to your specific field or industry. It may also be helpful to read reviews and compare different books before making a decision.
Yes, there are books that focus on specific types of optimisation, such as linear programming, nonlinear programming, and stochastic programming. It is important to choose a book that aligns with your specific needs and goals.
To apply the knowledge from a book on optimisation to real-world problems, it is important to understand the underlying principles and techniques, and then practice and adapt them to different scenarios. It may also be helpful to seek guidance from experts and collaborate with others in the field.