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- | ====== Mathematical Optimization ====== | ||
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- | Oh, Sertalp is handsome!FIXME | ||
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- | In mathematics, | ||
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- | 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" | ||
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- | ===== Optimization Problems ===== | ||
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- | An optimization problem can be represented in the following way: | ||
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- | > Given: a function $f : A \to R$ from some set $A$ to the real numbers | ||
- | > Sought: an element $x_0$ in $A$ such that $f(x_0) \leq f(x)$ for all $x$ in $A$ (" | ||
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- | Such a formulation is called an optimization problem or a mathematical programming problem (a term not directly related to computer programming, | ||
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- | Typically, $A$ is some subset of the Euclidean space $\mathbb{R}^n$, | ||
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- | By convention, the standard form of an optimization problem is 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, where a local minimum x* is defined as a point for which there exists some $\delta > 0$ 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. | ||
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- | ===== Notation ===== | ||
- | Optimization problems are often expressed with special notation. Here are some examples. | ||
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- | ==== Minimum and maximum value of a function ==== | ||
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- | Consider the following notation: | ||
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- | $$\min_{x\in\mathbb R}\; (x^2 + 1)$$ | ||
- | This denotes the minimum value of the objective function $x^2 + 1$, when choosing $x$ from the set of real numbers $\mathbb R$. The minimum value in this case is 1, occurring at $x = 0$. | ||
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- | Similarly, the notation | ||
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- | $$\max_{x\in\mathbb R}\; 2x$$ | ||
- | 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 " | ||
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- | ==== Optimal input arguments ==== | ||
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- | Consider the following notation: | ||
- | $$ | ||
- | \underset{x\in(-\infty, | ||
- | $$ | ||
- | or equivalently | ||
- | $$ | ||
- | \underset{x}{\operatorname{arg\, | ||
- | $$ | ||
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- | This represents the value (or values) of the argument $x$ in the interval $(-\infty, | ||
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- | Similarly, | ||
- | $$ | ||
- | \underset{x\in[-5, | ||
- | $$ | ||
- | or equivalently | ||
- | $$ | ||
- | \underset{x, | ||
- | $$ | ||
- | represents the $(x,y)$ pair (or pairs) that maximizes (or maximize) the value of the objective function $x\cos(y)$, with the added constraint that $x$ lie 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 $(5, 2k\pi)$ and $(−5, | ||
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- | **Arg min** and **arg max** are sometimes also written **argmin** and **argmax**, and stand for argument of the minimum and argument of the maximum. | ||