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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.
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.
Typical examples of global optimization applications include:
The most successful general strategies are:
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.
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 (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 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)
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
Several Monte-Carlo-based algorithms exist:
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.
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 (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, also known as replica exchange MCMC sampling, is a
Deterministic global optimization:
For simulated annealing:
For reactive search optimization:
For stochastic methods:
For parallel tempering:
For continuation methods:
For general considerations on the dimensionality of the domain of definition of the objective function:
2. Commercial:
1. Free and opensource:
Other approaches include heuristic strategies to search the search space in a more or less intelligent way, including:
Essentially, one runs N copies of the system, randomly initialized, at different temperatures. Then, based on the Metropolis criterion one exchanges configurations at different temperatures. The idea of this method is to make configurations at high temperatures available to the simulations at low temperatures and vice versa. This results in a very robust ensemble which is able to sample both low and high energy configurations. In this way, thermodynamical properties such as the specific heat, which is in general not well computed in the canonical ensemble, can be computed with great precision.
this is usually known as replica-exchange molecular dynamics or REMD. ^{[5]} version of parallel tempering:molecular dynamics Sugita and Okamoto formulated a ^{[4]}[3]
Computer science, Statistics, Discrete mathematics, Mathematical analysis, Interpolation
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