![]() The 'interior-point-legacy' method is based on LIPSOL (Linear Interior Point Solver,), which is a variant of Mehrotra's predictor-corrector algorithm, a primal-dual. IEEE/CAA Journal of Automatica Sinica publishes articles on original theoretical and experiment research and development in the area of automation science and.
Mathematical optimization - Wikipedia. Graph of a paraboloid given by z = f(x, y) = −(x² + y²) + 4. The global maximum at (x, y, z) = (0, 0, 4) is indicated by a blue dot. Nelder- Mead minimum search of Simionescu's function. Simplex vertices are ordered by their value, with 1 having the lowest (best) value. 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. The generalization of optimization theory and techniques to other formulations comprises a large area of applied mathematics. 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. Optimization problems[edit]An optimization problem can be represented in the following way: Given: a functionf : A→{\displaystyle \to }R from some set. A to the real numbers. Sought: an element x. A such that f(x. 0) ≤ f(x) for all x in A ("minimization") or such that f(x. 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 below). 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 or the choice set, while the elements of A are called candidate solutions or feasible solutions. The function f is called, variously, an objective function, a loss function or cost function (minimization),[2] a utility function or fitness function (maximization), or, in certain fields, an 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. In mathematics, conventional optimization problems are usually stated in terms of minimization. Generally, unless both the objective function and the feasible region are convex in a minimization problem, there may be several local minima. A local minimumx* is defined as a point for which there exists some δ > 0 such that for all x where∥x−x∗∥≤δ,{\displaystyle \|\mathbf {x} - \mathbf {x} ^{*}\|\leq \delta ,\,}the expressionf(x∗)≤f(x){\displaystyle 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. While a local minimum is at least as good as any nearby points, a global minimum is at least as good as every feasible point. In a convex problem, if there is a local minimum that is interior (not on the edge of the set of feasible points), 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. A 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. Notation[edit]Optimization problems are often expressed with special notation. Here are some examples. Minimum and maximum value of a function[edit]Consider the following notation: minx∈R(x. R} }\; (x^{2}+1)}This denotes the minimum value of the objective function x. R{\displaystyle \mathbb {R} }. The minimum value in this case is 1{\displaystyle 1}, occurring at x=0{\displaystyle x=0}. Similarly, the notationmaxx∈R2x{\displaystyle \max _{x\in \mathbb {R} }\; 2x}asks for the maximum value of the objective function 2x, where x may be any real number. In this case, there is no such maximum as the objective function is unbounded, so the answer is "infinity" or "undefined". Optimal input arguments[edit]Consider the following notation: argminx∈(−∞,−1]x. This represents the value (or values) of the argumentx in the interval(−∞,−1]{\displaystyle (- \infty ,- 1]} that minimizes (or minimize) the objective function x. In this case, the answer is x = –1, since x = 0 is infeasible, i. Similarly,argmaxx∈[−5,5],y∈Rxcos(y),{\displaystyle {\underset {x\in [- 5,5],\; y\in \mathbb {R} }{\operatorname {arg\,max} }}\; x\cos(y),}or equivalentlyargmaxx,yxcos(y),subject to: x∈[−5,5],y∈R,{\displaystyle {\underset {x,\; y}{\operatorname {arg\,max} }}\; x\cos(y),\; {\text{subject to: }}\; x\in [- 5,5],\; y\in \mathbb {R} ,}represents the (x,y){\displaystyle (x,y)} pair (or pairs) that maximizes (or maximize) the value of the objective function xcos(y){\displaystyle x\cos(y)}, with the added constraint that x lie in the interval [−5,5]{\displaystyle [- 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. History[edit]Fermat and Lagrange found calculus- based formulas for identifying optima, while Newton and Gauss proposed iterative methods for moving towards an optimum. The term "linear programming" for certain optimization cases was due to George B. Dantzig, although much of the theory had been introduced by Leonid Kantorovich in 1. 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 1. John von Neumann developed the theory of duality in the same year. Other major researchers in mathematical optimization include the following: Major subfields[edit]Convex programming studies the case when the objective function is convex (minimization) or concave (maximization) and the constraint set is convex. This can be viewed as a particular case of nonlinear programming or as generalization of linear or convex quadratic programming. Linear programming (LP), a type of convex programming, studies the case in which the objective function f is linear and the constraints are specified using only linear equalities and inequalities. Such a set is called a polyhedron or a polytope if it is bounded. Second order cone programming (SOCP) is a convex program, and includes certain types of quadratic programs. Semidefinite programming (SDP) is a subfield of convex optimization where the underlying variables are semidefinitematrices. It is a generalization of linear and convex quadratic programming. Conic programming is a general form of convex programming. LP, SOCP and SDP can all be viewed as conic programs with the appropriate type of cone. Geometric programming is a technique whereby objective and inequality constraints expressed as posynomials and equality constraints as monomials can be transformed into a convex program. Integer programming studies linear programs in which some or all variables are constrained to take on integer values. This is not convex, and in general much more difficult than regular linear programming. Quadratic programming allows the objective function to have quadratic terms, while the feasible set must be specified with linear equalities and inequalities. For specific forms of the quadratic term, this is a type of convex programming. Fractional programming studies optimization of ratios of two nonlinear functions. The special class of concave fractional programs can be transformed to a convex optimization problem.
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