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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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Optimization problem : Find value of variables such that error is mimimum

There are 3 variables x,y,z . x can have the integer values from 1-65535 , y can have integer values from 1-3 and z can have integer values from 0-2047. The equation is $$F = 13000000/((32x)*(y+(z/2048)))$$ F can take the values such as 4800 , 9600…
maheshin
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Critical points of absolute value function

For this question, if I divide this function into two parts, which are $x \ge 0$ and $x<0$, then the part that doesn't include "0" will have no critical point, and I also have no idea of how to find the singular point (might be the sharp point).…
Jakoer
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Finding all the minima in a unconstrained minimization problem

I've just read a textbook stating that the first order condition $\frac{df}{dx} = 0$ and second order sufficient condition $\frac{d^{2}f}{dx^2} \gt 0$ of unconstrained minimization will find the all the local minima. Is this statement right? Using…
user89855
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Minimize the trace of a combination of PSD matrices analytically

I have the following problem: Define $H$ and $R_k$ for $k=1\dots N$, to be $M\times M$ positive definite matrices. The problem is to find optimal weights $p_k$that solves the following problem \begin{equation*} \begin{aligned} &…
Rasheed
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Find max and min on some region

Find maximum and minimum values of $f(x,y,z)=x^2yz$ on the region $x^2+y^2\leq1,$ $0\leq z\leq1.$ First, I get $\nabla f= (2xyz, x^{2}z, x^{2}y) = (0,0,0) \implies x = y = z = 0$, so the critical point is $(0,0,0)$, and $f(0,0,0)=0$. There is no…
user133140
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Optimization of difference of two convex function.

Suppose we have to minimize a function $f(X,Y)-g(X,Y)$, where both the functions are convex. How can this be solved in matlab? Is there any tool to solve this in matlab?
user3256924
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Optimization - Get value of Lagrangian

We know that $f(x) \to \min$ subject to $g(x) = t$ and $h(x) \leq m$ can be written as $f(x) + \lambda g(x)\to\min$ subject to $h(x) \leq m$. How do we get value of lambda so that the two problems are equivalent.
Pratik Poddar
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Optimizing a value appearing in the Cheriton-Tarjan MST algorithm?

In the analysis of the Cheriton-Tarjan MST algortihm, there is a step that asks to optimize a quantity subject to a nonlinear constraint. Specifically, it's as follows: Let $c_1, c_2, ..., c_p$ be positive real numbers. Maximize…
templatetypedef
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Find the values of $c_1,c_2$ so that $(-0.5, 0, 0)$ is a point of a local maximum.

Given the problem $$ \max [-(x_1+x_2+x_3)]$$ subject to the contraints $$x_1^2+x_2^2=2c_1$$ $$x_1+5x_2+x_3^2=2c_2$$ I am asked to find the values of $c_1,c_2$ so that $(-0.5, 0, 0)$ is a point of a local maximum. I have found that $c_1=\frac{1}{8}$…
Mary Star
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How to define positive definite matrix?

why first matrix in fig is not positive definte, but second is a positive definite matrix. As Eigen values of both the matrix are non negative. and also real(A)-transpose(real(A)) is zero in both the cases. and also in both the diagonal elements…
user3256924
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Local minimizer

Given the feasible set, the point, and the gradient (and in some cases also the Hessian), how do you determine if the given point is definitely a local minimizer? Do you use the first-order necessary condition, and/or the second-order necessary…
user100503
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Please help me finding equilibrium output

Could anyone explain the step how to solve this problem?I guess equilibrium is same thing as profit maximization so I thought I need to know TR and TC first. I would like to solve this problem by myself but I really need someone's explanation…
emma
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Workflow Optimization

I originally posted the problem in stackoverflow but later on it become clear that it is more of a math problems then coding. Example: We have company doing support work on 3 projects (P1, P2, P3); In the company there are 4 workers (w1,w2,w3,w4)…
d.raev
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Maximizing the area of rectangle inscribed in triangle

I'd like to ask if someone could help me out with this problem. Let's have a triangle with coordinates $[0,0],[4,0],[1,3]$. Inscribe a rectangle into this triangle, so its area is maximized The base of rectangle lays on axis $x$. I know how to…
James
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First order necessary conditions for $\max_{x_1}f(x_1,g(x_1)).$

$$\max_{x_1}f(x_1,g(x_1)).$$ And, let $f$ attends max at $x_1^*$, so first order necessary conditions imply that $$\dfrac{\partial f(x_1^*,g(x_1^*))}{\partial x_1}+\dfrac{\partial f(x_1^*,g(x_1^*))}{\partial x_2}\dfrac{d g(x_1^*)}{dx_1}=0$$ and as…
Silent
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