Networks: Optimisation and Evolution (Cambridge Series in Statistical and Probabilistic Mathematics)

One of Fermat's theorems states that optima of unconstrained problems are found at stationary points , where the first derivative or the gradient of the objective function is zero see first derivative test. More generally, they may be found at critical points , where the first derivative or gradient of the objective function is zero or is undefined, or on the boundary of the choice set. An equation or set of equations stating that the first derivative s equal s zero at an interior optimum is called a 'first-order condition' or a set of first-order conditions.

Mathematical optimization

Optima of equality-constrained problems can be found by the Lagrange multiplier method. While the first derivative test identifies points that might be extrema, this test does not distinguish a point that is a minimum from one that is a maximum or one that is neither. When the objective function is twice differentiable, these cases can be distinguished by checking the second derivative or the matrix of second derivatives called the Hessian matrix in unconstrained problems, or the matrix of second derivatives of the objective function and the constraints called the bordered Hessian in constrained problems.

The conditions that distinguish maxima, or minima, from other stationary points are called 'second-order conditions' see ' Second derivative test '. If a candidate solution satisfies the first-order conditions, then satisfaction of the second-order conditions as well is sufficient to establish at least local optimality. The envelope theorem describes how the value of an optimal solution changes when an underlying parameter changes.

The process of computing this change is called comparative statics.


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The maximum theorem of Claude Berge describes the continuity of an optimal solution as a function of underlying parameters. For unconstrained problems with twice-differentiable functions, some critical points can be found by finding the points where the gradient of the objective function is zero that is, the stationary points.

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More generally, a zero subgradient certifies that a local minimum has been found for minimization problems with convex functions and other locally Lipschitz functions. Further, critical points can be classified using the definiteness of the Hessian matrix: If the Hessian is positive definite at a critical point, then the point is a local minimum; if the Hessian matrix is negative definite, then the point is a local maximum; finally, if indefinite, then the point is some kind of saddle point.

Constrained problems can often be transformed into unconstrained problems with the help of Lagrange multipliers. Lagrangian relaxation can also provide approximate solutions to difficult constrained problems.

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When the objective function is convex , then any local minimum will also be a global minimum. There exist efficient numerical techniques for minimizing convex functions, such as interior-point methods.

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To solve problems, researchers may use algorithms that terminate in a finite number of steps, or iterative methods that converge to a solution on some specified class of problems , or heuristics that may provide approximate solutions to some problems although their iterates need not converge. The iterative methods used to solve problems of nonlinear programming differ according to whether they evaluate Hessians , gradients, or only function values. While evaluating Hessians H and gradients G improves the rate of convergence, for functions for which these quantities exist and vary sufficiently smoothly, such evaluations increase the computational complexity or computational cost of each iteration.

In some cases, the computational complexity may be excessively high. One major criterion for optimizers is just the number of required function evaluations as this often is already a large computational effort, usually much more effort than within the optimizer itself, which mainly has to operate over the N variables. The derivatives provide detailed information for such optimizers, but are even harder to calculate, e.

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However, gradient optimizers need usually more iterations than Newton's algorithm. Which one is best with respect to the number of function calls depends on the problem itself. More generally, if the objective function is not a quadratic function, then many optimization methods use other methods to ensure that some subsequence of iterations converges to an optimal solution. The first and still popular method for ensuring convergence relies on line searches , which optimize a function along one dimension.

A second and increasingly popular method for ensuring convergence uses trust regions. Both line searches and trust regions are used in modern methods of non-differentiable optimization. Usually a global optimizer is much slower than advanced local optimizers such as BFGS , so often an efficient global optimizer can be constructed by starting the local optimizer from different starting points.

Besides finitely terminating algorithms and convergent iterative methods , there are heuristics. A heuristic is any algorithm which is not guaranteed mathematically to find the solution, but which is nevertheless useful in certain practical situations. List of some well-known heuristics:. 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 subset is the engineering optimization , and another 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.

This approach may be applied in cosmology and astrophysics,. Economics is closely enough linked to optimization of agents that an influential definition relatedly describes economics qua science as the "study of human behavior as a relationship between ends and scarce means" with alternative uses.

The Journal of Economic Literature codes classify mathematical programming, optimization techniques, and related topics under JEL: In microeconomics, the utility maximization problem and its dual problem , the expenditure minimization problem , are economic optimization problems. Insofar as they behave consistently, consumers are assumed to maximize their utility , while firms are usually assumed to maximize their profit.

Also, agents are often modeled as being risk-averse , thereby preferring to avoid risk. Asset prices are also modeled using optimization theory, though the underlying mathematics relies on optimizing stochastic processes rather than on static optimization. International trade theory also uses optimization to explain trade patterns between nations.

The optimization of portfolios is an example of multi-objective optimization in economics. Since the s, economists have modeled dynamic decisions over time using control theory. Some common applications of optimization techniques in electrical engineering include active filter design, [11] stray field reduction in superconducting magnetic energy storage systems, space mapping design of microwave structures, [12] handset antennas, [13] [14] [15] electromagnetics-based design.

Electromagnetically validated design optimization of microwave components and antennas has made extensive use of an appropriate physics-based or empirical surrogate model and space mapping methodologies since the discovery of space mapping in Optimization has been widely used in civil engineering. The most common civil engineering problems that are solved by optimization are cut and fill of roads, life-cycle analysis of structures and infrastructures, [18] resource leveling [19] and schedule optimization. Another field that uses optimization techniques extensively is operations research.

Increasingly, operations research uses stochastic programming to model dynamic decisions that adapt to events; such problems can be solved with large-scale optimization and stochastic optimization methods. Mathematical optimization is used in much modern controller design. High-level controllers such as model predictive control MPC or real-time optimization RTO employ mathematical optimization.

These algorithms run online and repeatedly determine values for decision variables, such as choke openings in a process plant, by iteratively solving a mathematical optimization problem including constraints and a model of the system to be controlled.

Optimization techniques are regularly used in geophysical parameter estimation problems. Given a set of geophysical measurements, e.


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Nonlinear optimization methods are widely used in conformational analysis. Methods to obtain suitable in some sense natural extensions of optimization problems that otherwise lack of existence or stability of solutions to obtain problems with guaranteed existence of solutions and their stability in some sense typically under various perturbation of data are in general called relaxation.

Relaxed problems may also possesses their own natural linear structure that may yield specific optimality conditions different from optimality conditions for the original problems. From Wikipedia, the free encyclopedia. For the peer-reviewed journal, see Mathematical Programming. For other uses, see Optimization disambiguation and Optimum disambiguation. Critical point mathematics , Differential calculus , Gradient , Hessian matrix , Positive definite matrix , Lipschitz continuity , Rademacher's theorem , Convex function , and Convex analysis. List of optimization algorithms.

Newton's method in optimization , Quasi-Newton method , Finite difference , Approximation theory , and Numerical analysis. Memetic algorithm Differential evolution Evolutionary algorithms Dynamic relaxation Genetic algorithms Hill climbing with random restart Nelder-Mead simplicial heuristic: List of optimization software. Brachistochrone Curve fitting Deterministic global optimization Goal programming Important publications in optimization Least squares Mathematical Optimization Society formerly Mathematical Programming Society Mathematical optimization algorithms Mathematical optimization software Process optimization Simulation-based optimization Test functions for optimization Variational calculus Vehicle routing problem.

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