Mathematical optimization

In mathematics, the simplest case of optimization, or mathematical programming, refers to the study of problems in which one seeks to minimize or maximize a real function by systematically choosing the values of real or integer variables from within an allowed set. This (a scalar real valued objective function) is actually a small subset of this field which comprises a large area of applied mathematics and generalizes to study of means to obtain "best available" values of some objective function given a defined domain where the elaboration is on the types of functions and the conditions and nature of the objects in the problem domain.

History

The first optimization technique, which is known as steepest descent, goes back to Gauss. Historically, the first term to be introduced was linear programming, which was invented by George Dantzig in the 1940s. The term programming in this context does not refer to computer programming (although computers are nowadays used extensively to solve mathematical problems). Instead, the term comes from the use of program by the United States military to refer to proposed training and logistics schedules, which were the problems that Dantzig was studying at the time. (Additionally, later on, the use of the term "programming" was apparently important for receiving government funding, as it was associated with high-technology research areas that were considered important.)

Other important mathematicians in the optimization field include:

• Richard Bellman
• Leonid Vitalyevich Kantorovich
• Leonid Khachian
• Arkadii Nemirovskii
• Yurii Nesterov
• John von Neumann
• Boris Polyak
• Naum Z. Shor
• Michael J. Todd
• Hoang Tuy
• Kees Roos (personal page: http://www.isa.ewi.tudelft.nl/~roos/)

Major subfields

• Linear programming studies the case in which the objective function f is linear and the set A is specified using only linear equalities and inequalities. Such a set is called a polyhedron or a polytope if it is bounded.
• Integer programming studies linear programs in which some or all variables are constrained to take on integer values.
• Quadratic programming allows the objective function to have quadratic terms, while the set A must be specified with linear equalities and inequalities.
• Nonlinear programming studies the general case in which the objective function or the constraints or both contain nonlinear parts.
• Convex programming studies the case when the objective function is convex and the constraints, if any, form a convex set. This can be viewed as a particular case of nonlinear programming or as generalization of linear or convex quadratic programming.
  • Second order cone programming (SOCP).
• Semidefinite programming (SDP) is a subfield of convex optimization where the underlying variables are semidefinite matrices. It is generalization of linear and convex quadratic programming.
• Stochastic programming studies the case in which some of the constraints or parameters depends on random variables.
• Robust programming is, as stochastic programming, an attempt to capture uncertainty in the data underlying the optimization problem. This is not done through the use of random variables, but instead, the problem is solved taking into account inaccuracies in the input data.
• Combinatorial optimization is concerned with problems where the set of feasible solutions is discrete or can be reduced to a discrete one.
• Infinite-dimensional optimization studies the case when the set of feasible solutions is a subset of an infinite-dimensional space, such as a space of functions.
• Heuristic algorithms
  • Metaheuristics
• Constraint satisfaction studies the case in which the objective function f is constant (this is used in artificial intelligence, particularly in automated reasoning).
  • Constraint programming.
• Disjunctive programming used where at least one constraint must be satisfied but not all. Of particular use in scheduling.
• Trajectory optimization is the speciality of optimizing trajectories for air and space vehicles.

In a number of subfields, the techniques are designed primarily for optimization in dynamic contexts (that is, decision making over time):

• Calculus of variations seeks to optimize an objective defined over many points in time, by considering how the objective function changes if there is a small change in the choice path.
• Optimal control theory is a generalization of the calculus of variations.
• Dynamic programming studies the case in which the optimization strategy is based on splitting the problem into smaller subproblems. The equation that relates these subproblems is called the Bellman equation.
• Mathematical programming with equilibrium constraints is where the constraints include variational inequalities or complementarities.

Optimization topics

Optimization problems

An optimization problem can be represented in the following way

Given: a function f : A $\to$ R from some set A to the real numbers

Sought: an element x₀ in A such that f(x₀) ≤ f(x) for all x in A ("minimization") or such that f(x₀) ≥ f(x) for all x in A ("maximization").

Such a formulation is called an optimization problem or a mathematical programming problem (a term not directly related to computer programming, but still in use for example in linear programming - see History above). Many real-world and theoretical problems may be modeled in this general framework. Problems formulated using this technique in the fields of physics and computer vision may refer to the technique as energy minimization, speaking of the value of the function f as representing the energy of the system being modeled.

Typically, A is some subset of the Euclidean space Rⁿ, often specified by a set of constraints, equalities or inequalities that the members of A have to satisfy. The domain A of f is called the search space, while the elements of A are called candidate solutions or feasible solutions.

The function f is called, variously, an objective function, cost function, energy function, or energy functional. A feasible solution that minimizes (or maximizes, if that is the goal) the objective function is called an optimal solution.

Generally, when the feasible region or the objective function of the problem does not present convexity, there may be several local minima and maxima, where a local minimum x^* is defined as a point for which there exists some δ > 0 so that for all x such that

$\|\mathbf {x} -\mathbf {x} ^{*}\|\leq \delta ;\,$

the expression

$f(\mathbf {x} ^{*})\leq f(\mathbf {x} )$

holds; that is to say, on some region around x^* all of the function values are greater than or equal to the value at that point. Local maxima are defined similarly.

A large number of algorithms proposed for solving non-convex problems – including the majority of commercially available solvers – are not capable of making a distinction between local optimal solutions and rigorous optimal solutions, and will treat the former as actual solutions to the original problem. The branch of applied mathematics and numerical analysis that is concerned with the development of deterministic algorithms that are capable of guaranteeing convergence in finite time to the actual optimal solution of a non-convex problem is called global optimization.

Notation

Optimization problems are often expressed with special notation. Here are some examples.

$\min _{x\in \mathbb {R} }\;(x^{2}+1)$

This asks for the minimum value for the objective function $x$²$+1$, where $x$ ranges over the real numbers $\mathbb {R}$. The minimum value in this case is $1$, occurring at $x=0$.

$\max _{x\in \mathbb {R} }\;2x$

This asks for the maximum value for the objective function 2x, where x ranges over the reals. In this case, there is no such maximum as the objective function is unbounded, so the answer is "infinity" or "undefined".

$\operatorname {argmin} _{x\in [-\infty ;-1]}\;x^{2}+1\,$

This asks for the value (or values) of x in the interval $[-\infty ;-1]$ that minimizes (or minimize) the objective function x² + 1 (the actual minimum value of that function does not matter). In this case, the answer is $x$ = -1.

$\operatorname {argmax} _{x\in [-5,5],\;y\in \mathbb {R} }\;x\cdot \cos(y)\,$

This asks for the $(x,y)$ pair (or pairs) that maximizes (or maximize) the value of the objective function $x*cos(y)$, with the added constraint that $x$ lies in the interval $[-5;5]$ (again, the actual maximum value of the expression does not matter). In this case, the solutions are the pairs of the form ( 5 ; 2kπ ) and ( −5 ;(2k+1)π ), where $k$ ranges over all integers.

Techniques

Crudely all the methods are divided according to variables called:-
SVO:- Single Variable Optimization
MVO:- Multi Variable Optimization
For twice-differentiable functions, unconstrained problems can be solved by finding the points where the gradient of the objective function is zero (that is, the stationary points) and using the Hessian matrix to classify the type of each point. If the Hessian is positive definite, the point is a local minimum, if negative definite, a local maximum, and if indefinite it is some kind of saddle point.

The existence of derivatives is not always assumed and many methods were devised for specific situations. The basic classes of methods, based on smoothness of the objective function, are:

• Combinatorial methods
• Derivative-free methods
• First order methods
• Second-order methods

Actual methods falling somewhere among the categories above include:

• Bundle methods
• Conjugate gradient method
• Ellipsoid method
• Gradient descent aka steepest descent or steepest ascent
• Interior point methods
• Line search - a technique for one dimensional optimization, usually used as a subroutine for other, more general techniques.
• Nelder-Mead method aka the Amoeba method
• Newton's method
• Quasi-Newton methods
• Simplex method
• Subgradient method - similar to gradient method in case there are no gradients

Should the objective function be convex over the region of interest, then any local minimum will also be a global minimum. There exist robust, fast numerical techniques for optimizing twice differentiable convex functions. This is for your concern.

Constrained problems can often be transformed into unconstrained problems with the help of Lagrange multipliers.

Here are a few other popular methods:

• Ant colony optimization
• Beam search
• Bees algorithm
• Differential evolution
• Dynamic relaxation
• Evolution strategy
• Genetic algorithms
• Harmony search
• Hill climbing
• Particle swarm optimization
• Quantum annealing
• Simulated annealing
• Stochastic tunneling
• Tabu search

Multi-objective optimization

Main article: Multiobjective optimization

Adding more than one objective to an optimization problem adds complexity. For example, if you wanted to optimize a structural design, you would want a design that is both light and rigid. Because these two objectives conflict, a trade-off exists. There will be one lightest design, one stiffest design, and an infinite number of designs that are some compromise of weight and stiffness. This set of trade-off designs is known as a Pareto set. The curve created plotting weight against stiffness of the best designs is known as the Pareto frontier.

A design is judged to be Pareto optimal if it is not dominated by other designs: a Pareto optimal design must be better than another design in at least one aspect. If it is worse than another design in all respects, then it is dominated and is not Pareto optimal.

Uses

Problems in rigid body dynamics (in particular articulated rigid body dynamics) often require mathematical programming techniques, since you can view rigid body dynamics as attempting to solve an ordinary differential equation on a constraint manifold; the constraints are various nonlinear geometric constraints such as "these two points must always coincide", "this surface must not penetrate any other", or "this point must always lie somewhere on this curve". Also, the problem of computing contact forces can be done by solving a linear complementarity problem, which can also be viewed as a QP (quadratic programming) problem.

Many design problems can also be expressed as optimization programs. This application is called design optimization. One recent and growing subset of this field is multidisciplinary design optimization, which, while useful in many problems, has in particular been applied to aerospace engineering problems.

Mainstream economics also relies heavily on mathematical programming. An often studied problem in microeconomics, the utility maximization problem, and its dual problem the Expenditure minimization problem, are economic optimization problems. Consumers and firms are assumed to maximize their utility/profit. Also, agents are most frequently assumed to be risk-averse thereby wishing to minimize whatever risk they might be exposed to. Asset prices are also explained using optimization though the underlying theory is more complicated than simple utility or profit optimization. Trade theory also uses optimization to explain trade patterns between nations.

Another field that uses optimization techniques extensively is operations research.

See also

• Arg max
• Brachistochrone
• Dynamic programming
• Fuzzy logic
• Game theory
• Goal Programming

• Important publications in optimization
• Interior point methods
• Operations research
• Optimization problem
• Mathematical optimization software
• Process optimization

• Radial basis function
• Random optimization
• Simplex algorithm
• Topkis's Theorem
• Variational calculus
• Variational inequality

Solvers

• CPLEX
• COMET is a language dedicated to optimization: constraint-based local search, constraint-programming, mathematical programming (free for academic use).
• IMSL Numerical Libraries are collections of math and statistical algorithms available in C/C++, Fortran, Java and C#/.NET. Optimization routines in the IMSL Libraries include unconstrained, linearly and nonlinearly constrained minimizations, and linear programming algorithms.
• IPOPT - an open-source primal-dual interior point method NLP solver which handles sparse matrices
• KNITRO - solver for nonlinear optimization problems
• Mathematica - handles linear programming, integer programming and constrained non-linear optimization problems
• NAG Numerical Libraries-The NAG Library contains a comprehensive collection of Optimization routines, which cover a diverse set of problems and circumstances.http://www.nag.co.uk/optimization/index.asp
• OpenOpt - a free toolbox with connections to lots of solvers, for Python language programmers
• Merlin - A Fortran-77, user friendly open source software package, for non-linear optimization with bound constraints. Available from the URL: http://merlin.cs.uoi.gr

References

• Mordecai Avriel (2003). Nonlinear Programming: Analysis and Methods. Dover Publishing. ISBN 0-486-43227-0.
• Stephen Boyd and Lieven Vandenberghe (2004). Convex Optimization, Cambridge University Press. ISBN 0-521-83378-7.
• Elster K.-H. (1993), Modern Mathematical Methods of Optimization, Vch Pub. ISBN 3055014529.
• Jorge Nocedal and Stephen J. Wright (2006). Numerical Optimization, Springer. ISBN 0-387-30303-0.
• Panos Y. Papalambros and Douglass J. Wilde (2000). Principles of Optimal Design : Modeling and Computation, Cambridge University Press. ISBN 0-521-62727-3.
• Yang X.-S. (2008), Introduction to Mathematical Optimization: From Linear Programming to Metaheuristics, Cambridge Int. Science Publishing. ISBN 1-904602827.

External links

• COIN-OR — Computational Infrastructure for Operations Research
• Convex Optimization & Euclidean Distance Geometry by Jon Dattorro
• Decision Tree for Optimization Software Links to optimization source codes
• Global optimization
• Mathematical Programming Glossary
• Mathematical optimization
• Mathematical Programming Society
• NEOS Guide currently being replaced by the NEOS Wiki
• Optimization Online A repository for optimization e-prints
• Optimization Related Links
• The Basics of Practical Optimization An undergraduate optimization text
• Convex Optimization I EE364a: Course from Stanford University
• Convex Optimization – Boyd and Vandenberghe Book on Convex Optimization
• Simplemax Online Optimization Services Web applications to access nonlinear optimization services

Modeling languages:

• AIMMS
• AMPL
• GAMS — General Algebraic Modeling System
• MPL
• OPL
• OPTIMJ

Solvers:

• CONOPT
• CPLEX - linear, quadratic, and mixed-integer programming solver
• Free Optimization Software by Systems Optimization Laboratory, Stanford University
• Gurobi - parallel linear and mixed-integer programming solver
• JOpt
• Moocho - a very flexible open-source NLP solver
• Mosek - linear, quadratic, conic and mixed-integer programming solver
• SAS/OR
• SmartDO - Engineering global optimization (commercial) software
• TANGO Project - Trustable Algorithms for Nonlinear General Optimization

Libraries:

• ALGLIB Optimization sources. C++, C#, Delphi, Visual Basic.
• CPLEX Component Libraries
• IOptLib (Investigative Optimization Library) - a free open source library for development of optimization algorithms (ANSI C).
• OAT (Optimization Algorithm Toolkit) - a set of standard optimization algorithms and problems in Java.
• OOL (Open Optimization library) - a set of optimization routines in C.