 #jsDisabledContent { display:none; } My Account |  Register |  Help Flag as Inappropriate This article will be permanently flagged as inappropriate and made unaccessible to everyone. Are you certain this article is inappropriate?          Excessive Violence          Sexual Content          Political / Social Email this Article Email Address:

# Global optimization

Article Id: WHEBN0000563854
Reproduction Date:

 Title: Global optimization Author: World Heritage Encyclopedia Language: English Subject: Collection: Mathematical Optimization Publisher: World Heritage Encyclopedia Publication Date:

### Global optimization

Global optimization is a branch of applied mathematics and numerical analysis that deals with the global optimization of a function or a set of functions according to some criteria. Typically, a set of bound and more general constraints is also present, and the decision variables are optimized considering also the constraints.

Global optimization is distinguished from regular optimization by its focus on finding the maximum or minimum over all input values, as opposed to finding local minima or maxima.

## Contents

• General 1
• Applications 2
• Deterministic methods 3
• Inner and outer approximation 3.1
• Cutting plane methods 3.2
• Branch and bound methods 3.3
• Interval methods 3.4
• Methods based on real algebraic geometry 3.5
• Stochastic methods 4
• Simulated annealing 4.1
• Direct Monte-Carlo sampling 4.2
• Stochastic tunneling 4.3
• Parallel tempering 4.4
• Heuristics and metaheuristics 5
• Response surface methodology based approaches 6
• Software 7
• Footnotes 9
• References 10

## General

A common (standard) model form is the minimization of one real-valued function f in the parameter-space \vec{x}\in P, or its specified subset \vec{x}\in D: here D denotes the set defined by the constraints.

(The maximization of a real-valued function g(x) is equivalent to the minimization of the function f(x):=(-1)\cdot g(x).)

In many nonlinear optimization problems, the objective function f has a large number of local minima and maxima. Finding an arbitrary local optimum is relatively straightforward by using classical local optimization methods. Finding the global minimum (or maximum) of a function is far more difficult: symbolic (analytical) methods are frequently not applicable, and the use of numerical solution strategies often leads to very hard challenges.

## Applications

Typical examples of global optimization applications include:

## Deterministic methods

The most successful general strategies are:

### Inner and outer approximation

In both of these strategies, set over which a function is to be optimized is approximated by polyhedra. In inner approximation, the polyhedra are contained in the set, while in outer approximation, the polyhedra contain the set.

### Cutting plane methods

The cutting-plane method is an umbrella term for optimization methods which iteratively refine a feasible set or objective function by means of linear inequalities, termed cuts. Such procedures are popularly used to find integer solutions to mixed integer linear programming (MILP) problems, as well as to solve general, not necessarily differentiable convex optimization problems. The use of cutting planes to solve MILP was introduced by Ralph E. Gomory and Václav Chvátal.

### Branch and bound methods

Branch and bound (BB or B&B) is an algorithm design paradigm for discrete and combinatorial optimization problems. 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. The algorithm explores branches of this tree, which represent subsets of the solution set. Before enumerating the candidate solutions of a branch, the branch is checked against upper and lower estimated bounds on the optimal solution, and is discarded if it cannot produce a better solution than the best one found so far by the algorithm.

### Interval methods

Interval arithmetic, interval mathematics, interval analysis, or interval computation, is a method developed by mathematicians since the 1950s and 1960s as an approach to putting bounds on rounding errors and measurement errors in mathematical computation and thus developing numerical methods that yield reliable results. Interval arithmetic helps find reliable and guaranteed solutions to equations and optimization problems.

(see interalg from OpenOpt and GlobSol)

### Methods based on real algebraic geometry

Real algebra is the part of algebra which is relevant to real algebraic (and semialgebraic) geometry. It is mostly concerned with the study of ordered fields and ordered rings (in particular real closed fields) and their applications to the study of positive polynomials and sums-of-squares of polynomials. It can be used in convex optimization

## Stochastic methods

Several Monte-Carlo-based algorithms exist:

### Simulated annealing

Simulated annealing (SA)' is a generic probabilistic metaheuristic for the global optimization problem of locating a good approximation to the global optimum of a given function in a large search space. It is often used when the search space is discrete (e.g., all tours that visit a given set of cities). For certain problems, simulated annealing may be more efficient than exhaustive enumeration — provided that the goal is merely to find an acceptably good solution in a fixed amount of time, rather than the best possible solution.

### Direct Monte-Carlo sampling

In this method, random simulations are used to find an approximate solution.

Example: The traveling salesman problem is what is called a conventional optimization problem. That is, all the facts (distances between each destination point) needed to determine the optimal path to follow are known with certainty and the goal is to run through the possible travel choices to come up with the one with the lowest total distance. However, let's assume that instead of wanting to minimize the total distance traveled to visit each desired destination, we wanted to minimize the total time needed to reach each destination. This goes beyond conventional optimization since travel time is inherently uncertain (traffic jams, time of day, etc.). As a result, to determine our optimal path we would want to use simulation - optimization to first understand the range of potential times it could take to go from one point to another (represented by a probability distribution in this case rather than a specific distance) and then optimize our travel decisions to identify the best path to follow taking that uncertainty into account.

### Stochastic tunneling

Stochastic tunneling (STUN) is an approach to global optimization based on the Monte Carlo method-sampling of the function to be objective minimized in which the function is nonlinearly transformed to allow for easier tunneling among regions containing function minima. Easier tunneling allows for faster exploration of sample space and faster convergence to a good solution.

### Parallel tempering

Parallel tempering, also known as replica exchange MCMC sampling, is a

• A. Neumaier’s page on Global Optimization
• Introduction to global optimization by L. Liberti
• Free e-book by Thomas Weise