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Mathematical optimization

In mathematics, computer science and operations research, mathematical optimization or mathematical programming, alternatively spelled optimisation, is the selection of a best element (with regard to some criterion) from some set of available alternatives.[1]

In the simplest case, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function.

More generally, optimization includes finding 'best available' values of some objective function given a defined domain (or input), including a variety of different types of objective functions and different types of domains.

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 below).

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

n

a utility function or fitness function (maximization), or, in certain fields, an energy function or energy functional.

x

∗

x

∗

a convex problem, if there is a local minimum that is interior (not on the edge of the set of feasible elements), it is also the global minimum, but a nonconvex problem may have more than one local minimum not all of which need be global minima.

large number of algorithms proposed for solving nonconvex problems – including the majority of commercially available solvers – are not capable of making a distinction between locally optimal solutions and globally optimal solutions, and will treat the former as actual solutions to the original problem.

Global optimization is 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 nonconvex problem.

2

2

(Programming in this context does not refer to computer programming, but from the use of program by the United States military to refer to proposed training and logistics schedules, which were the problems Dantzig studied at that time.) Dantzig published the Simplex algorithm in 1947, and John von Neumann developed the theory of duality in the same year.

design is judged to be 'Pareto optimal' (equivalently, 'Pareto efficient' or in the Pareto set) if it is not dominated by any other design: If it is worse than another design in some respects and no better in any respect, then it is dominated and is not Pareto optimal.

In other words, defining the problem as multi-objective optimization signals that some information is missing: desirable objectives are given but combinations of them are not rated relative to each other.

Multi-objective optimization problems have been generalized further into vector optimization problems where the (partial) ordering is no longer given by the Pareto ordering.

Classical optimization techniques due to their iterative approach do not perform satisfactorily when they are used to obtain multiple solutions, since it is not guaranteed that different solutions will be obtained even with different starting points in multiple runs of the algorithm.

The satisfiability problem, also called the feasibility problem, is just the problem of finding any feasible solution at all without regard to objective value.

The extreme value theorem of Karl Weierstrass states that a continuous real-valued function on a compact set attains its maximum and minimum value.

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.

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.

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.

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).

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;

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).

Introduction An optimization algorithm is a procedure which is executed iteratively by comparing various solutions till an optimum or a satisfactory solution will be found.

Optimization algorithms help us to minimize or maximize an objective function E(x) which is simply a mathematical function dependent on the Model’s internal parameters which are used in computing the target values(Y) from the set of predictors(X) used in the model.

Now we all know a Neural Network trains via a famous technique called Back-propagation , in which propagating forward calculating the dot product of Inputs signals and their corresponding Weights and then applying a activation function to those sum of products, which transforms the input signal to an output signal and also is important to model complex Non-linear functions and introduces Non-linearity to the Model which enables the Model to learn almost any arbitrary functional mapping.[8]

The second order derivative informs us whether the first derivative is increasing or decreasing which hints at the function’s curvature.It also provides us with a quadratic surface which touches the curvature of the Error Surface.[10]

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.

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

Newton's method requires the 2nd order derivates, so for each iteration the number of function calls is in the order of N², but for a simpler pure gradient optimizer it is only N.

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.

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.

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[12];

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.

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.[14]

Modern optimization theory includes traditional optimization theory but also overlaps with game theory and the study of economic equilibria.

Asset prices are also modeled using optimization theory, though the underlying mathematics relies on optimizing stochastic processes rather than on static optimization.

Macroeconomists build dynamic stochastic general equilibrium (DSGE) models that describe the dynamics of the whole economy as the result of the interdependent optimizing decisions of workers, consumers, investors, and governments.[18][19]

stray field reduction in superconducting magnetic energy storage systems, space mapping design of microwave structures,[21]

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 1993.[25][26]

The most common civil engineering problems that are solved by optimization are cut and fill of roads, life-cycle analysis of structures and infrastructures,[27]

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 used in many facets of computational systems biology such as model building, optimal experimental design, metabolic engineering, and synthetic biology.[30]

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.

Unsatisfied Clause

GSAT and WalkSAT algorithms can be generalised to weighted MAX-SAT by using the objective function for weighted MAX-SAT – that is, the total weight of the clauses unsatisfied under a given assignment – as the evaluation function based on which the variable to be flipped in each search step is selected.

When hard constraints are explicitly identified (via a lower bound on the weights of CNF clauses that are to be treated as hard constraints), this WalkSAT algorithm restricts the clause selection in the first stage of the variable selection mechanism to unsatisfied hard constraint clauses, unless all hard constraints are satisfied by the current candidate assignment.

The motivation behind the latter mechanism is based on the following observations: In situations where many clauses are unsatisfied, the probability for selecting the best clause, that is, the unsatisfied clause that contains one of the variables whose flip leads to a maximal improvement in the objective function value, can be very small when basing this selection on a uniform distribution, as used in standard WalkSAT.

However, for various types of MAX-SAT-encoded instances of other problems, including well-known classes of minimum-cost graph colouring and set covering instances, Novelty+/wcs+we appears to find quasi-optimal (i.e., best known) solutions in significantly less CPU time than other high-performance algorithms for MAX-SAT, such as IRoTS or GLS, and appears to be the best-performing MAX-SAT algorithm known to date [Hoos et al., 2003].

Satisfiability Problem

The Maximum Satisfiability Problem (MAX-SAT) is the optimisation variant of SAT in which the goal is to find a variable assignment that maximises the number or total weight of satisfied clauses.

For widely used types of benchmark instances, including test-sets of randomly generated MAX-SAT instances as well as MAX-SAT encodings of other combinatorial optimisation problems (such as set covering and time-tabling), these approximation algorithms do not reach the performance of even relatively simple SLS algorithms.

Some of these algorithms achieve state-of-the-art performance on mildly overconstrained instances whose optimal solutions leave relatively few clauses unsatisfied (GLS as well as Novelty+ and its variants for weighted MAX-SAT seem to fall into this category), while others, such as IRoTS, appear to be state-of-the-art for highly overconstrained instances.

to the number of literals that are simultaneously satisfied in a given clause, leads to theoretical and practical improvements in the performance of simple iterative improvement methods for unweighted MAX-SAT, there is little evidence that non-oblivious evaluation functions are instrumental in reaching state-of-the-art SLS performance on MAX-SAT instances of any type.

On the other hand, generalisations of WalkSAT to overconstrained pseudo-Boolean CSP and integer programming problems, which can be seen as special cases of MAX-CSP, have been used successfully to solve various application problems, and in many cases, they have been shown to achieve substantially better performance than specialised algorithms and state-of-the-art commercial optimisation tools.

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