# Optimization

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Optimization problems usually feature three fundamental elements: one numerical quantity, or objective function that can be maximized or minimized. The second element is a collection of variables. The third element of any optimization problem is constraints on the values the variables can receive. The development of optimization techniques paralleled advances not only in computer science but also in operational analysis, numerical analysis, game theory, mathematics economics control theory, and combinators. Other useful classes of optimization difficulties not covered in this article include stochastic programming when the objective function or constraint is dependent on random variables.

## Mathematical optimization

In mathematical programming, the objective function and the constraints are mathematical formulas that relate a set of variables over some domain. The process by which a mathematical problem is formulated as an optimization problem is known as modelling.

Mathematical optimization shares many techniques with numerical analysis, a branch of applied mathematics. To find or approximate solutions to optimization problems, Algorithms are used for modelling, reducing the problem into one or more computational problems, and then solving these numerical problems.

Optimization is a standard term in business. Businesses use it to find the best possible outcome in what they do will produce themselves, whether it's making food or engineering a building, materials are used most efficiently.

According to the business dictionary, optimization is improving the quality of something by storing historical data and comparing it with current situations. Businesses depend on this crucial information to make decisions regarding their product or services.

Optimization techniques are applied in many fields of study including artificial intelligence, management science, computer science, engineering, and operations research. In computer science, an optimization algorithm is a specific type of algorithm that modifies the code of another program to improve one or more properties such as speed or memory usage.

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## Optimization problems

Optimization problems feature three elements: index variables of some domain, objective functions, and constraints on these variables. The first element of every optimization problem is the index variables of some domain. Index variables are numerical quantities that can change depending on changes in other variables through an equation or function.

Objective functions are numerical quantities that one wishes to maximize or minimize over some domain. Any solution to an optimization problem must have at least one optimal value for the objective function that it tries to maximize or minimize.

Constraints on these variables ensure that they still fit within whatever bounds may be required. For example, if you are shooting a basketball from a point on the ground and at a particular height, you want to know the angle that makes it go into the hoop.

Math and science problems feature three elements: index variables of some domain, objective functions, and constraints on these variables. The first element in every optimization problem is the index variables for some domain. Index variables are numerical quantities that can change depending on changes in other variables through an equation or function.

Objective functions are numerical quantities that one wishes to maximize or minimize over some domain. Any solution must have at least one optimal value for the objective function it tries to maximize or minimize.

Constraints on this variable ensure that they still fit within whatever bounds are required for them. For example, if you're shooting a basketball from a point on the ground and at a particular height, you want to know the angle that makes it go into the hoop.

## History

The history of optimization traces the historical development of a phenomenon that is fundamental to all applied sciences. Its origins can be traced back to antiquity, but it was not until the Renaissance that mathematics and physics became linked together through geometry. In his book" History of Mathematical Notations"

In 1630 Descartes discovered another general method in his "La Geometrie". He reduced geometric problems to algebraic equations by a process he called analytic geometry. ... Later, Newton and Leibniz introduced a more sophisticated application of mathematical reasoning in their attempts to solve the old problem of quadratures involved in measuring areas and volumes.

The first attempt to find the length of an unknown curve was done by Johann Bernoulli in 1696. His formula, however, does not converge for any continuous curve other than the Archimedean spiral.

Bernoulli's principle provided a method for determining both tangential and normal components of acceleration when the velocity is known. To do this he assumed that ax + b = c where x is the distance along the path, a is acceleration, and b and c are constants.

## Computational optimization techniques

The history of computational optimization techniques follows fairly closely the same path as that for mathematical optimization, albeit with a much later start.

In 1827-1828 Jacobi and Steiner developed the important idea of a function approximation by polynomials. Bernhard Riemann published in 1851 his insight into the size of error bounds for approximating integrals using finite differences. In particular, he gave conditions for an algorithm to be considered "uniformly convergent" (a basic notion still used today).

Then in 1870 Heine derived lower bounds on errors caused by interpolating functions that are not polynomial (today's most popular type of approximation functions), thus changing the situation fundamentally.

Bertrand provided another powerful tool with his theorem on the approximation of functions by polynomials, which was important for both computational and theoretical work. This amounted to an early version of the modern notion of a "subgradient" function.

The mid-twentieth century saw two related events that changed dramatically our ability to solve real-life optimization problems: The introduction in 1947 by George Dantzig of linear programming (LP) together with his demonstration that it could be used to efficiently solve many practical problems; and, at about the same time, Karush's seminal idea (in conjunction with his student Dantzig) concerning the reduction of general nonlinear optimization problems to LP.

While LP has now become part of standard mathematics curricula around the world, is used to solve a myriad of problems in industry, and has been extensively analyzed and refined by many researchers, it is Karush's basic idea that forms the basis for today's global optimization community.

The development of economic models together with linear programming had given rise to operational research (OR) [1]. Its origins go back to about 1940 when the term was coined by the founders of a group at M.I.T., but its scope grew slowly until about 1960 when two things happened: First, OR became established as a distinct field within applied mathematics; second, it became clear that OR needed new mathematical tools permitting better problems definition and solution techniques.

Among these developments was activity in multi-objective linear programming which matured into what later became known as multi-objective optimization.

In the late 1950s and early 1960s, problems from a wide range of disciplines started being solved using new techniques which, in retrospect, can be viewed as having laid the foundations for today's global optimization.

In 1959 an important change became evident with the introduction of trust-region (TR) methods that considerably reduced the number of function evaluations needed to solve convex problems. This led later to further developments like a barrier or augmented Lagrangian descent methods known today as a sequential quadratic programming (SQP).

SQP has resulted in significant advances in solving both smooth and nonsmooth nonlinear programs (NLP), now widely used by industry. All these methods are now considered classical, yet they enjoy considerable interest in their application to problems arising in areas such as finance, computational biology, chemistry, geosciences, and other fields.

Today's global optimization community is built on these foundations through the efforts of thousands of researchers who have contributed over the past half-century to solving many real-life problems using efficient algorithms.

It has led to a vast literature that is now accessible for the first time thanks to the World Wide Web (WWW). The present volume aims at being an up-to-date snapshot of this state-of-the-art field which is both challenging and rewarding for its practitioners.

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The volume contains about 30 contributions written by recognized experts in this particular field; most articles include exercises and references which may serve as starting points

What are the major subfields of global optimization

Though there is no consensus on this topic, most researchers would probably mention at least one of the following sub-disciplines:

· convex and nonconvex programming;

· unconstrained and constrained optimization;

· continuous and discrete optimization;

· smooth and nonsmooth optimization;

· global optimal solutions (optimality criteria);

· integer programming (IP);

· efficient algorithms for solving IP problems.

What are the most interesting open research problems in global optimization?

What do you see as possible future directions or new frontiers in your field?

There is certainly still much room for improvement of both algorithms and problem formulations. On the theoretical side, one can mention in particular:

· further development of penetration and trust-region approaches to solving NLP with special emphasis on nonsmooth problems;

· understanding global optimization algorithms in terms of their convergence properties in particular for smooth problems when the number of function evaluations "m" approaches infinity.

· improving existing methods for solving IP that is based on lowering or linearizing the barrier/Lagrangian either through primal or dual coordinate pivoting strategies.

This is probably still an open research area since many new methodologies will be required at both the algorithmic (algebraic geometry) and problem formulation (geometric programming) levels;

· investigating alternative formulations of the IP problem.

On the application side we can mention:

· developing new objective functions for open-pit mine planning and allocation problems;

· solving optimization problems over Riemannian manifolds such as the operational space of a blast furnace, which opens up research on mathematical programming in noncommutative spaces;

· investigating approaches to perform global optimization when only local information is available (e.g., constraints and/or data points);

· studying optimal problem formulations (including bounds and constraints), which is particularly relevant concerning high-dimensional systems;

· improving methods for modelling uncertain parameters. What do you consider interesting open problems? What needs to be done that has not been done yet?

We consider that global optimization can still offer new perspectives given the following points. First, it offers an approach to solve NP-hard problems, which appear to be ubiquitous in many fields (e.g., scheduling, logistics, circuits design). Second, thanks to interior-point methods for solving NLP arising from convex programming or related sub-disciplines coming from continuous optimization (Lagrangian relaxation) and discrete optimization (gradient projection algorithms), it allows one to build powerful hybrid algorithms with promising computational performances; third, by relying on machine learning techniques like artificial neural networks or genetic algorithms, global optimization is well suited for designing robust control laws for complex systems. Last but not least, global optimization leads naturally to relax of nonconvex objectives, which is still an important topic in many fields.

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