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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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Supplier Costs Minimization

I have a warehouse. The customer makes some orders and I have to make a correspondent order to the supplier. Than when the item arrives I will dispatch the item to the customer. I would optimize the process, e.g. minimizing the shipping costs. For…
gdm
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Minimize a $C^1$ function subject to convex constraint set using feasible directions.

Let $X$ be a convex set. Let $f:\mathbb{R}^n \to \mathbb{R}$ a $C^1$ function. The idea is to minimize f subject to $x \in X$. Let $x^* \in X$ be a point such that $$\nabla f(x^*)^T (x-x^*) \geq 0$$ for all $x \in X$. Then can I conclude that $x^*$…
root
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Equivalence of two optimization problem when introducing an auxiliary variable

Consider two optimization problems: $$ (P1:) \ \ \min\limits_{x \in X} x $$ $$ (P2:) \ \ \min\limits_{x \in X, x \le y} y $$ Many authors said that P1 and P2 are equivalent, but without any further explanation. In my opinion, P1 can be seen as…
Fengsheng Wei
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Solving a quadratic convex optimization problem

The optimization problem that I am solving is of the form min ||x||^2 (square of l-2 norm of x) such that Mx <= b. Here x is a d-dimensional vector, M is the transpose of a d-dimensional vector and b is a scalar. I started this by writing the…
Aditi Gupta
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Minimize $f(x)$ subject to a constraint

I'm stuck on a problem for my optimization class. Minimize $f(x)$ st $x \in \Omega$ where $\Omega=\{x\in \mathbb{R^2}:x_1^2+x_2^2\geq 1\}$ and $ f(x)=x_2$. a) Find all points that satisfy FONC b) Which points from (a) satisfy SONC c) Which points…
Albibi
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Which books to choose for optimization?

I'm considering to choose between "Convex Optimization" by Stephen Boyd and Lieven Vandenberghe, and Yurii Nesterov's "Lectures on Convex Optimization" to supplement for my university course. I have had Calculus and Linear Algebra, my course covers…
alksdhalksjdb
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"If $x^*$ is a local maximum, then $f''(x^*)$ is less than zero." Why is this false?

Since $\,f''(x^*)<0$, implies local max. Can you give me examples where $x^*$ is a local max but $\,f''(x^*)=0$.
Joa
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How can we show the equivalence of two optimizations?

How can we show the equivalence of (a) and (b)? $\max(x^TAx)$ subject to $\|x\|=1$....(a) $\max{\frac{x^TAx}{x^Tx}}$...(b)
DSPinfinity
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find coefficients that maximize an objective function

I have an object function, $Obj(t)$, that I want to minimize. I have $1000$ test cases that returns different outcomes for $t$. The test cases can return different outcome even with the same $t$ value. Each of $1000$ test cases have input variables…
emerson
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Analytical solution for minimax optimization problem

In my personal studies/exploration I have come across the following problem: $$\inf_{a \geq 0} \sup_{\substack{0 \leq z \leq 1\\t > 0}} z\sqrt{b(a^2 + s^2)} - \dfrac{at}{2} - \dfrac{az^2}{2}\left(\dfrac{1}{a\sqrt{\delta} + t}\right) + …
user539287
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Showing that v(S), the linear variety generated by subspace S, is a linear variety

The other question related to this section of the book appears to be unresolved; I just started Luenberger's Optimization by Vector Space Methods and have reached a point of confusion. Two definitions given: A linear variety $V$ of a subspace $M$…
lazyhippo
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Linear Optimization - Best algorithm?

I have multiple sensors of the same kind that measure tempearture. The temperature it returns is actually in a digital (byte) format. The manufacturer have provided us with a linear equation (with coefficients m and b) to convert the digital bytes…
muosac
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Optimization - different objective same constraints

I have two optimization problems $min G(x), Ax=b$ and $min F(x), Ax=b$. Assume continuity, differentiability, convexity of $F(x)$ and $G(x)$. What is the relationship between optimal solutions of $F(x)$ and $G(x)$ ?
bissi
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Are $\min_{x,y} f(x,y)$ and $\min_{x} \min_{y} f(x,y)$ equivalent?

Are the following two equations equivalent? The first equation is $$ \min_{x,y} f(x,y). $$ The second equation is $$ \min_x g(x) \quad \text{where} \quad g(x) = \min_y f(x,y). $$ The second equation can be considered as $$\min_x \left(\min_y…
Danny_Kim
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Maximizing a product of real numbers under a constraint

Good evening, I was solving an interesting problem from IMO 1976 : Determine the greatest number, who is the product of some positive integers, and the sum of these numbers is 1976. And I was wondering how could we solve it if it was a product of…
LexLarn
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