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Mathematics Stack Exchange

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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how to interpret local minima of combinatorial optimization

I am having a difficult time trying to interpret and visualize the local minima of a combinatorial optimization objective function. Here's a rough sketch of my problem: I have $m$ points $x_1,\dots,x_m$ in $\mathbb{R}^d$. My objective function is…
Tomas Jorovic
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Graphical solution (with two variables), solution properties.

(c) infeasibility depends on the constraints; if we look at the graph we can see that (1, 1) is the intersection of constraint (I) and (II), and for this to be infeasible we need t in constraint (III) to $t>3$, because then if you won't find a…
Dick Armstrong
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Inspecting a project network

From the given table one can draw the project/activity network above. There are four possible paths: (i) C, B. (ii) C, A, D. (iii) E, D. (iv) G, F, D. The first path's project time is 4+5=9, the second path's project time is 4+2+9=15, the third…
Dick Armstrong
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Best basis selection problem with inequalities solution methods

I'm solving an optimization problem in form $\min \sum x$ subject to, $ A x = b$ (1) $ g x \leq d$ (2) $ x \geq 0$ (3) Optimization variable is x. The number of rows of A is considerably higher than the number of columns, i.e., the system $Ax =…
Premysl Vorac
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Given model determine Least Squares estimate - is my graph correct?

Given the model $\alpha_1 I_k - \alpha_2 I_k^2 = I_{k+1}$ corresponding to measurements shown in the table below $k$ | 1 | 2 | 3 | 4 $I_k$| 0 | 1 | 10 | 80 | Determine the least squares estimate of $\alpha_1$ and $\alpha_2$ I got $A =…
Egbert
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Optimization problem: maximise horizontal distance

Fifteen men are placed on a Dead Man's Chest in a rectangular pattern, with each man distant $a$ from his neighbours,thus: The average weight of the men is $w$, and the heaviest man weighs no more than $2w$. Find the maximum possible horizontal…
Rescy_
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Is the given function convex? Does the point satisfy the FONC?

minimize $2x_1^2+x_2^2-2x_1x_2-4x_2$ subject to $x_1x_2\leq 4$ Is the objective function convex? Does the point $(1,4)$ satisfy the FONC for a local minimizer? Do optimal solutions of this given problem change if the constraint is replaced by the…
Alice
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Fit polynomial function using experimental data (least squares)

I want to fit the polynomial function $f(x) = \alpha_0 +\alpha_1 x +\alpha_2 x^2 $ using given data such that the errors $y_c-f(x_c)$ are minimized (least squares). Obtained is the experimental data shown below. $c $|-2 -1 0 1 2 $x_c|$-2 -1 0 1…
Steve
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Find vector that maximizes $f(x) = 2x_1^2+2x_2^2-x_3^3+2x_1x_2$

Find the vector with $||x||^2=x^Tx=1$ that maximizes the following function. $f(x) = 2x_1^2+2x_2^2-x_3^3+2x_1x_2$ I have rewritten the quadratic form as $f(x) = \frac{1}{2}x^T \begin{bmatrix} 4&2&0\\2&4&0\\0&0&-2\end{bmatrix} x$ in which the…
Peter
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Significance of lower semicontinuity in (non-)convex optimisation

In the context of (non-)convex optimisation, what is the reason behind requiring that the objective function be lower semicontinuous? From what I understand, 1) a function is continuous iff it is both lower and upper semicontinuous, and 2) a convex…
John Lee
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What is the solution of this optimization problem?

I am looking for the solution of this optimization problem: $$ \min_{x} \max_{1 \leq r\leq N-1} \left|\frac{\sin\pi r M x}{\sin\pi r x}\right|^2$$ where $M \ll N$, $x \in \mathbb{R}$, $r \in \mathbb{N}$. My attempts: For $M = 2$ it can be shown that…
Mahdi Khosravi
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KKT multipliers sign convention

We all know that if we have an optimization problem of the general form: $$ \min f(x) $$ subject to: $$\begin{align} h(x) &= 0 \\ g(x) &\le 0. \end{align}$$ Then, we have a vector of multipliers, let's call it $s$, associated with the equality…
user254769
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Is there any method that convert a concave problem into convex problem?

I have an optimization problem of the form: \begin{align} \begin{cases} x_2 \rightarrow \min, \\ \text{subject to:} \\ f_1(x) \leq 0, \\ f_2(x) \leq 0, \end{cases} \end{align} with $x= (x_1,x_2)^T$ as the optimization variable. Here, $f_1(\cdot)$ is…
Tina
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maximization and minimization

Show that $f(x_1^*,...x_n^*)=\max\{f(x_1,...,x_n):(x_1,...,x_n)\in\Omega\}$ if and only if $-f(x_1^*,...x_n^*)=\min\{-f(x_1,...,x_n):(x_1,...,x_n)\in\Omega\}$ I am not exactly sure how to approach this problem -- it is very general, so I can't…
Emir
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Asset optimization using Excel

I originally posted this in superuser, but was told it was more of a math problem than an Excel problem. I'm working on a project, part of which involves figuring out the number and type of vessels required to transport x amount of crude oil over a…
youjustreadthis
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