Lecture

Optimization: Mathematical Principles and Algorithms

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Description

This lecture introduces the main mathematical principles and important algorithms of optimization, using examples like the positioning of a marble in a box and the computation of optimal itineraries in complex networks. The instructor will guide students through linear and nonlinear optimization problems, suggesting the implementation of algorithms in Python within Jupyter Notebook to enhance understanding.

In MOOCs (3)

Optimization: principles and algorithms - Network and discrete optimization

Introduction to the mathematical concept of networks, and to two important optimization problems on networks: the transshipment problem and the shortest path problem. Short introduction to the modeling power of discrete optimization, with reference to classical problems. Introduction to the branch and bound algorithm, and the concept of cuts.

Optimization: principles and algorithms - Unconstrained nonlinear optimization

What you'll learn Formulation: you will learn from simple examples how to formulate, transform and characterize an optimization problem. Objective function: you will review the mathematical properties of the objective function that are important in optimization. Optimality conditions: you will learn sufficient and necessary conditions for an optimal solution. Solving equations, Newton: this is a reminder about Newton's method to solve nonlinear equations. Newton's local method: you will see how to interpret

Optimization: principles and algorithms - Linear optimization

What you'll learn Formulation: you will learn from simple examples how to formulate, transform and characterize an optimization problem. Constraints: you will learn how to represent the constraints of a linear optimization problem, both from a geometric and algebraic point of views. Duality: you will learn how to derive a companion problem called the "dual". Optimality conditions: you will learn sufficient and necessary conditions for an optimal solution. Simplex method: you will learn an algorithm to solve

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Related concepts (32)

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.

Convex optimization

Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex optimization problems admit polynomial-time algorithms, whereas mathematical optimization is in general NP-hard.

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.

Multi-objective optimization

Multi-objective optimization or Pareto optimization (also known as multi-objective programming, vector optimization, multicriteria optimization, or multiattribute optimization) is an area of multiple-criteria decision making that is concerned with mathematical optimization problems involving more than one objective function to be optimized simultaneously. Multi-objective is a type of vector optimization that has been applied in many fields of science, including engineering, economics and logistics where optimal decisions need to be taken in the presence of trade-offs between two or more conflicting objectives.

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