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.
23,419 questions
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Maximum Travelling Salesman Problem on the integer line
I am trying to solve the following problem:
Let's say a "frog" is jumping on the numberline starting at $0$, jumps randomly on every integer from $1,\dots,n$ and then comes back to 0. What ...
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Maximisation of functions of the form $f(x) = \sqrt{1 - x^2} + (ax+b)x$
I am studying a function arising in the analysis of robust aggregation rules in distributed learning, but the question is purely analytical. The function I am facing depends on parameters $a, b > 0$...
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Generalization of "Neighbours in a matrix"
Lately, I was doing problems on the book "The Art of Mathematics : Coffee time in Memphis".
One certain problem caught my attention, the problem $21$, Neighbors in a matrix. Basically it ...
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Simplex Method - How to know which basis to choose after the first iteration?
I'm trying to understand how to correctly choose the next basis after the first iteration in the simplex method.
In my problem, I have the following minimization form:
$$
\begin{aligned}
\min z &= ...
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Maximizing the area of a triangle
I am given 3 radii $r_a, r_b, r_c$ and I want to determine the 3 angles $\phi_a,\phi_b,\phi_c$ for which the area of the triangle defined by $\left(r_a\cos(\phi_a),r_a\sin(\phi_a)\right),\,\left(r_b\...
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Fermat-Torricelli Weighted Point
I am in the process of designing a global trajectory program for civil aircraft. Two aircraft depart from their airports, join together to create a formation, then later separate and land at their ...
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Linearization Question for max-min|x| Bi-level Optimization Problem
I'm currently working on a bi-level optimization problem with the following structure:
max min |x|
I attempted to linearize this problem using the following approach:
Introduce an auxiliary variable ...
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How to see $x_1\geq 0$ in the primal implies $p' A_1 \leq c_1$ in the dual, when there are no other constraints than just $x_1\geq 0$?
I understand why the constraint $x_1 \geq 0$ in the primal, implies $p'A_1 \leq c_1$ in the dual in the presence of a constraint involving an $A$ and a vector $b$ in the primal ($A_1$ being the first ...
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When does an optimization problem that is "worth solving" also satisfy strong duality?
I want some minimalistic/easy to remember statements that relate someone's willingness to solve an optimization problem with the strong duality property of that problem. Of course, you would only want ...
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Solving for a diffraction grating's orientation and surface periodicity from its diffracted beams, can the problem be inverted?
Background:
Below is a calculation of the diffracted wavevector $\mathbf{k_1}$ from an initial wavevector $\mathbf{k_0} = -k_0 \mathbf{\hat{z}}$ incident on a planar surface with normal $\mathbf{\hat{...
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Max–min assignment on a DAG when nodes have candidate values with compatibility constraints
I have a DAG where every node has a (usually small) set of candidate integers. A candidate a is compatible with b if (a | b) or (b | a). For every root I want to choose one candidate per node to ...
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Does block coordinate descent lead to a constrained critical point?
Let $f:\mathbb{R}^N\to[0, \infty)$ be a non-convex function of $w\in\mathbb{R}^N$ for $N\in\mathbb{N}$. Suppose the entries in $w$ can be partitioned into two vectors $u\in\mathbb{R}^m$ and $v\in\...
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How many big and small marbles are not used?
Translating to English from a non-English physics book about measurements:
Anif has $8$ big marbles and $15$ small marbles. The weight of the big and small marbles are $37.5$ and $12.5$ respectively. ...
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Optimal strategy for combinatorial marble-drawing game
You have $a$ amber, $b$ bronze, and $c$ crimson colored marbles in your hand, with $a\geq b\geq c$. An exact copy of this set of marbles is in a bag. Every turn, you select a marble from your hand to ...
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It seems that the definition of dual problem in Boyd & Vandenberghe does not apply to SVM
In Section 5.2 of Boyd & Vandenberghe's Convex Optimization, the dual problem given a primal problem with only inequality constraint is,
$$ \max \quad g(\alpha) \\ \text{ s. t.} \quad \alpha \...
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