Mathematics Stack Exchange
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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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Solving optimization problem (shrinkage)

I am trying to solve the optimization problem $$\underset{x}{\arg\min}\left\{||x||_2+\frac{\beta}{2}||x-v||_2^2\right\}$$ where $\beta>0$ and $v$ is some fixed vector. $\textbf{My approach so far:}$ I know that $||x||_2$ is not differentiable at…
James
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Simple Optimization problem single variable

Very abstractly suppose that the number of tokens I buy depends on the formula $Y_{input} = \sqrt{200000x} - 10 000$ the amount that goes out for the same $x$ somewhere else is $Y_{output} = 16 853 - \frac{500 000}{50.15-\sqrt{\frac{200000}{x}}}$ In…
qubitz
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Take too long time for solving MIQP

I'm solving the mixed-integer quadratic problem with CPLEX. The objective, constraints, and decision variable sizes are following: Objective matrix: $Q \in R^{312 \times 312}$ Ineqality matrix: $A_{ineq} \in R^{480 \times 312}$ Eqaulity matrix:…
hyeong seok Lee
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Thresholding for relaxed linear constraint to satisfy the combinatorial constraint

I have to satisfy a combinatorial constraint in $y$ with $y_i \in \{0,1\}$ and $i = \{1,2,..,n\}$, $\sum_i y_i \leq k$. The process of obtaining y is as follows Sample $x \in [0,1]^n$ Project x onto half space $\sum_i x_i \leq c_1$. (based on $L_1$…
NKJ
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Stationary point without a positive semi-definite hessian matrix cannot be a local minimizer?

I understand that if a hessian is positive semidefinite for all x, the function is convex (though not strictly convex), so the stationary point is a local and global minimum. However, what if a stationary point does not have a positive semi-definite…
Jayden Rice
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Knight covering the whole grid?

A knight (the chess piece) $m$ x $n$ grid Start from any square Question: For which, general case, $m$ x $n$ (neglecting the obvious failes $m,n=1,2$) is this possible/impossible without visiting the same square more than once? Can you…
Prim3numbah
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Paying $n$ people such that the ratio of their pays matches their quality ratio and subject to minimum wage constraints

You are trying to pay $n$ people. Each person $i \in \{1, \ldots, n\}$ has an expected minimum wage $w_i > 0$ and a quality $q_i > 0$. Denote the payment for each person as $p_i$. We require: (1) Each person must be paid at least their expected…
24n8
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Can't find the roots for this function even though there is a global/local maximum, why?

I'm trying to find the roots for this function: $$ f'(x) =…
Hiperfly
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How to approach this sum-minimization problem

I am new to math. How to approach the following problem? $\min_{a,b} \sum_{t=1}^N (-4aX_t\sin(Z_tb) -4aY_t\cos(Z_tb)+a^2Z_t^2 + X_t^2 + Y_t^2)$ where $X_t,Y_t,Z_t$ for $t\in \{1,...N\}$ are given. Say $N$ is around 800. Do I need software? Which? I…
u17
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When is it not a good idea to substitute the constraint into the objective function?

If I maximize $x^3/3-3y^2/2+2x \;$ subject to $x-y=0$ by using the Lagrangean method and confirm the bordered Hessian condition, I get that no solution exists. This can also be seen by looking at the graph. However, if I substitute the constraint…
PGupta
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Solving optimization problem by penalty method

Given this minimization problem: $$ \text{minimize }\, \, x_1^2 + 2x_2^2 \\ \text{subject to } \, \, x_1 + x_2 = 3 $$ I wish to solve this using the penalty method, what I've done so far: $$ \text{minimize} \, \, f(x) \\ \text{where} \, \, f(x) =…
Bob Pen
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Maximum points in a rowing and planking contest

I am trying to solve a problem that was asked by a friend of mine who is attending high school. As it has something to do with linear optimization, I think I should be more than capable of solving the problem as I got really good results from my…
Mathias
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Is it possible to say that the number of summation in the objective of optimizations affect the speed of solving them?

I have two different optimization models to solve a problem. Both of them work on a graph but the first one leads to optimization with an objective that has two summations: $\sum \sum x_{ij}$ and the second one leads to optimization with one…
samie
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how to minimise difference of data?

I have a data $X(i)=i$, where $i \in ${1,2,3,4,5,6}, i want to find $u$ that minimises $max|X(i)-u|$. How to solve this problem, can you give a proof?
anfjnlnl
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Fastest time between two points by walking under some constraints

So the problem, is that I want to walk A to B under the fastest time. If I had no constraints, then the answer is simple, simply a straight line from A to B but here I have some constraint which makes it complicated. Here are my rules and set…
tryst with freedom
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