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Questions tagged [optimization]

Optimization is the process of choosing the "best" value among possible values. They are often formulated as questions on the minimization/maximization of functions, with or without constraints.

In mathematics, computer science, economics, or management science, mathematical optimization (alternatively, optimization or mathematical programming) is the selection of a best element (with regard to some criteria) from some set of available alternatives.

An optimization problem can be represented in the following way: given a function $f:A\to\mathbb{R}$ from some set $A$ to the real numbers, we want to find an element $x_0\in A$ such that $f(x_0)\le f(x)$ for all $x \in A$ ("minimization") or such that $f(x_0)\ge f(x)$ for all $x \in A$ ("maximization").

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Derivative of Lagrangian does not depend on x?

Consider the following problem: $$ \min \mathbf{c}^T\mathbf{x}, \mathbf{x} \in \mathbf{R}^N\\ \mathbf{A}\mathbf{x}=\mathbf{b} $$ where $\mathbf{A}\in\mathbf{R}^{D \times N}, \mathbf{b}\in\mathbf{R}^D$. I am trying to find the Lagrangian and then the…
Joe Doe
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Normalization of Two Conflicting Objectives for Optimization

I'm writing a computer routine in which I'm trying to optimize two objectives (cost and utility). Since the two objectives are in conflict, I would like to normalize the values of the two functions and add them together to reach a compromize. What I…
PPGoodMan
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Does a concave/convex real valued function with a nonempty, convex and compact domain has a maximizer?

Let $X$ be a nonempty, convex and compact subset of $\mathbb{R}$ and $f:X\rightarrow \mathbb{R}$ be a real-valued function. Does $f$ always has a maximizer in $X$ if it is concave? Does $f$ always has a maximizer in $X$ if it is convex? Since $X$ is…
I000
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How to show that maximal subset is same as sum over maximal subset

Consider a collection of positive values, $x_i$, $i=1,\ldots,n$. Is it true that the maximum $k
Rong Zhang
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Optimization: Solving the problem $\min_{x \in \mathbb{R}^n} \sum_{k=1}^{m} a_k \|x-x_k\|^2$

I do not know how to start with the following exercise: Let $x_1,x_2,\ldots,x_m \in \mathbb{R}^n$ be arbitrary and $a_1,a_2,\ldots,a_m$ strictly positive real numbers. Solve the following problem: $$\min_{x \in \mathbb{R}^n} \sum_{k=1}^{m} a_k…
3nondatur
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Is $\min_x f(x) + \min_{y} g(x,y) = \min_{x,y} f(x) + g(x,y)?$

Do we in general have $$\min_x \left[f(x) + \min_{y} g(x,y) \right]= \min_{x,y} f(x) + g(x,y),$$ i.e. we choose some $x$ which gives us an $f(x)$, and then, treating $x$ as constant, we minimize $g(x,y)$ in $y$. Is this equivalent to choosing $x, y$…
Oeao
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Minimize the number of elements to complete the maximum amount of sets.

Consider a set of distinct elements $\{a_1, a_2, a_3,\dots, a_n\}$ (the universe) and a collection of sets $S= \{ S_1, S_2, S_3,\dots, S_m \} $ with each set formed out of any strictly positive number of elements. For a given number $k$ with $0
KooDooMoo
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Can we have same cut?

In Benders decomposition algorithm, we first solve a master problem which provides us with a solution vector. We then use this vector to construct a subproblem. For a minimization problem, subproblems give an upper bound . if different between upper…
ken kavaza
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Determine the largest possible number of square plots into which she can divide a field

A fenced, rectangular meadow has dimensions $24$ m by $52$ m. A mathematical farmer has $2019$ m of fence that he wants to use to fence the outside of the field and to partition the field into identical square plots with sides parallel to the edges…
Andy
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Optimization over multiple variables with a single objective

I'm dealing with an optimization problem of the form $$\underset{x,y}{\text{min}}\ f(x,y)$$ but I'm having trouble finding a good reference on how to deal with this. Through Google searches, I think I've found that a problem like this can be…
rbell
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Is this an error in the Quasi-Newton condition slide?

The confusing part is at this point in the lecture from the Numerical Optimization course. Consider a simple way to update $B^k$: Let $a \neq 0, u \in R, u \neq 0$ $$B^{k+1} = B^k + \alpha uu^T$$ Choose a and u such that B^{k+1} satisfies the…
Marko Grdinić
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What is the main characteristics of separable and non-separable functions?

I have read an article that says, from the optimization point of view, the separable functions are simpler then non-separable functions and optimization algorithms can find their optimal values more easily. What are the main characteristics of…
Optimized Life
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Confusion related to the convexity of norms

I am a bit confused about the proof of the convexity of the norm. Suppose I consider 1 norm form in which case the function is $f(x) = ||x||_1$ Referring to this pdf, http://ttic.uchicago.edu/~argyriou/courses/lec3.pdf. They have given something…
user34790
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Maximization with lin-log objective

Let $N = \{1,\ldots,n\}$. For a given $(r_1, \ldots, r_n) \in \mathbb R_{++}^n$. I need to solve \begin{align} \max_{(k_1, \ldots, k_n)\in \mathbb R_{++}^n} \prod_{i \in N}{\left[\sum_{j \in N}{(r_j - k_j)} +…
clueless
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Monotonic Transformation and Optimization

Let us consider the following function $f:X\rightarrow{R}$. Let us denote by $f_{max}$ and $f_{min}$, respectively, maximum and minimum values of $f$ function. Let us consider the following monotonic function $g:X\rightarrow{R}$. By using monotonic…
David
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