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.

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De Jong's Fifth Function's Minimum?

What is the minimum solution to De Jong's fifth function, in the range $-65.536\leqslant x_1\leqslant 65.536, -65.536\leqslant x_2\leqslant 65.536$?
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The minimum value of $\frac{a(x+a)^2}{\sqrt{x^2-a^2}}$

The problem is to find the minimum of $A$, which I attempted and got a different answer than my book: $$A=\frac{a(x+a)^2}{\sqrt{x^2-a^2}}$$ where $a$ is a constant ...
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3answers
446 views

Intuition behind gradient VS curvature

In Newton's method, one computes the gradient of a cost function, (the 'slope') as well as its hessian matrix, (ie, second derivative of the cost function, or 'curvature'). I understand the intuition, ...
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1answer
28 views

What does 'the level set is bounded' exactly want to tell?

'The level set is bounded.' occurs in many theorems and other places. I think I can understand the definition of 'level set' but I don't know what does 'it's bounded' want to tell me exactly in ...
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37 views

Solving for f(t) in presence of f'(t)

Here's the situation: I have a function $$e(t) = \frac{a~d(t)}{b + d(t)}$$ with first derivative $$e'(t) = \frac{a~b~d'(t)}{[b+d(t)]^2}$$ where $a$ and $b$ are constants. For a given constant $K$ I ...
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23 views

Maximize area of a corral

See problem 7 and figure 9 in picture What I've done so far: Not sure if $P=2l+2w$ or just $l+2w$ (dashed line makes me think the latter) $600=\pi r+l+2w$ $600=\pi r+2r+2w$ $w=\frac{600-\pi ...
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38 views

Shannon Entropy Continuity Constraint

I have the following problem: I want to find the probability density $p$ which maximizes the Shannon entropy \begin{equation} S := - \int_{x_b}^{x_c} dx ~ p(x) \log (p(x)) \end{equation} under the ...
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31 views

What is this sort of optimisation called?

I am reading a book in mathematical finance. There is something about constrained optimisation. They have specialised it for the financial market, but I am wondering what the general name for this ...
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9 views

Strong duality and its relation with perturbation functions

From the strong duality wiki page Strong duality holds if $F^{**}=F$ where $F$ is the perturbation function relating the primal and dual problems and $F^{**}$ is the biconjugate of $F$. I ...
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392 views

Optimizing a functional with a differential equation as a constraint

I am working on solving the following optimization problem. I think it is well-poised but, if not, please give me some pointers that could make the question make more sense. We have a parametric ...
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1answer
33 views

Given an optimal solution to the LP, show how it can be used to construct a directed cycle with minimal directed cycle mean cost.

Let $\mathcal G = (\mathcal V, \mathcal A)$ be directed graph with associated edge costs $c_{i,j}$ that has at least one directed cycle. Define the directed cycle mean cost to be $\frac {\{\text {sum ...
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74 views

A interesting max min problem

Let $\mathcal{S}\subset\mathbb{R}^2$ be a bounded, closed, compact, convex set which contains origin in its interior. Define \begin{align} c_1^{\star}=\min_{{(x_1,0)\in\mathcal{S}}}~&x_1 ...
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28 views

Split a vector into three

Say we have a vector of length n<100, $v(w_1,w_2,\ldots,w_{n})$. My problem is to divide the vector $v$ into groups of $3$, eg $u_m =(w_i, w_k, w_k)$ with as close weight as possible. Eg to ...
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877 views

How to describe minimization of L1 norm error using linear programming?

Given a set of $n$ pair points $(x_1, y_1), ..., (x_n, y_n)$ in the plane, I need to find a line $ax + by = c$ that fits the points of the L1 norm error points as closely as possible. I need a linear ...
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9 views

Binary Linear Programm: Check for feasability and multiple solutions

Assuming, I have binary integer program, e.g. given by: $ \arg\min_x \quad 0\\ \text{such that}\quad A_\text{eq} x = b_\text{eq}, x_i \in \{0,1\} $ Where also $[A_\text{eq}]_{ij} \in \{0,1\} $ and ...
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22 views

Property of monotone operator (Positive definite)

I would like to prove this statement: "$F$ is monotone if and only if $\nabla F$ is positive semidefinte." I only know $F$ is monotone with respect to $\Omega$ if and only if ...
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2answers
256 views

why in Phase I of the simplex method, if artificial variable become nonbasic, it never become basic?

Does anybody has idea how to solve this problem ? "Show that in Phase I of the simplex method, if an articial variable becomes nonbasic, it need never again become basic. Thus, when an articial ...
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35 views

How to show these two problems have equivalent solutions

I have two problems, where $A$ is positive definite: $$\inf \{ x^TAx + b^Tx : 1-x^Tx \ge 0,\ x \in \mathbb R^n\} \ (1)$$ and $$ max_\lambda \ q(\lambda) = -0.25b^T(A+\lambda I)^{-1}b - \lambda : ...
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32 views

How can I show that these two problems have the same optimal solution?

How can I show that these two problems have the same optimal solution: $$\inf \{ x^TAx + b^Tx : 1-x^Tx \ge 0,\ x \in \mathbb R^n\}$$ $$\inf \{ x^TAx + b^Tx : 1-x^Tx = 0,\ x \in \mathbb R^n\}$$ when ...
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521 views

Notation: what is “arg min”

there is a function that says j=arg_min{ f(x),g(y) } What does that mean? As noted by the comment, arg_min(f(x)) is the x that gives smalles f. But what happens when arg_min takes two function, ...
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Discretization of a convolution integral for constrained optimization problem

I'm working on a constrained optimization problem in which an unknown forcing function, $u(\eta)$, is in the integrand of a convolution integral. To find an optimal shape for $u(\eta)$, the integral ...
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Optimization problem with embedded absolute values (how to turn to LP)

say I have a problem of the form $$\begin{align*}\min&\sum_i{c_i\,|x_i-|y_i-z_i|+|s_i|\,|}\\[0.3cm] \text{s.t. }&0\le x\le 1\\[0.2cm] &0\le y \le 1\\[0.2cm] &0\le z\le 1\\[0.2cm] ...
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478 views

How to find the minimum/maximum distance of a point from elipse

I have the point $(1,-1)$ and the ellipse $$x^2/9 + y^2/5 = 1 $$ How to find the minimum and maximum distance of the point from the ellipse ? from exploring the ellipse I know that $$a = 3$$ , $$b ...
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2answers
29 views

Given a fixed volume for a solid cylinder, is it possible to find the minimum or maximum curved surface area?

Given a fixed volume for a solid cylinder, is it possible to find the minimum or maximum curved surface area $ 2 \pi r h $ of this cylinder?
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32 views

How to get the peak value of this logarithmic equation?

Is there a way to get the peak point of the following equation? $$ (a_1-a_2 x)\ln\left(1+\frac{b_1 x}{b_2 x+b_3}\right),$$ where $a_1,a_2,b_1,b_2,b_3$ are all positive constant values and $x$ is also ...
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33 views

Local optimality of a KKT point.

Consider the problem \begin{equation} \min_x f(x)~~~{\rm s.t.}~~~ g_i(x)\leq 0,~~i=1,\dots,I, \end{equation} where $x$ is the optimization parameter vector, $f(x)$ is the objective function and ...
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47 views

Prove an artificial variable that leaves the basis will never return.

Prove an artificial variable that leaves the basis will never return. Edit: This is for the simplex method (I think). I have no idea how to start this. Anyone know any books with these kinds of ...
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Find those values 'a' which belongs to the Convex Hull

Find those values of 'a' for which (1,a,1) belongs to the convex hull of $$\{(0,0,0), (1,1,2),(2,4,-6), (1,3,8)\}$$ Give me hints as much as you can, I would like to understand the mindset rather ...
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Solving this in order to L: (6x-L)(6y-L)-9=0

I don't really know how to explain this in english since I study it in portuguese, but I can't get my head around to solve this. My book says the solution to this problem is L1=6x V L2=36x^2 * y^2 -9 ...
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2answers
544 views

Optimal Basic Feasible Solutions

In linear programming, is it true that you can only have at most 2 optimal basic feasible solutions? If so, why?
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Optimizing a set of rules to better predict the outcome of events

I'm trying to better predict the top three finishers of the next 1000 800m mens freestyle swimming race. I've got a set of rules to rate the swimmers: 1) Add 5 points if the swimmer won his last ...
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One solution of a diophantine system

How to find one solution of $Ax = b$, where $A$ is a $(m, n)$ matrix and $x$ a vector of size $(n, 1)$. $A$, $x$ and $b$ are matrices of integers entries. How to check whether is a solution exists?
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32 views

Conditions for unique solution of a maximization problem?

Let $S\subseteq \mathbb{R}^2$, $d:=(d_1,d_2) \in S$, and $s:=(s_1,s_2)$ a generic point of $S$. Assume that there exists $s \in S$ such that $s_1>d_1$ and $s_2 >d_2$. Consider the following ...
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A Special Case of Maximum Coverage Problem

Let us denote $[N]=\{1,2,\cdots,\ N\}$ and let $\{a_{ij}\}_{i,j\in [N]}$ are positive numbers. Then how to proceed to solve the following problem $$\max_{G\subset[N]:|G|\le K}\left(\sum_{i\in ...
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382 views

Moore–Penrose pseudoinverse reference

Given the eigendecompositions $AA^{\top}=Q \Lambda Q^{\top}$ and $A^{\top}A=P \Lambda P^{\top}$, where $\Lambda$ is a diagonal matrix (of eigenvalues) and $P$ and $Q$ are unitary eigenvectors matrices ...
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Question about norm of diference of minimum of functions

let's say you have a function $f(x,y):\mathbb{R}^n \times \mathbb{R}^n \mapsto \mathbb{R}$ which is differntiable with respect to $x$ and has (at least) a minimizer on that variable. Then define ...
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Solving for stationary points for questions of the following type

How do you solve questions like $f(x,y) = x^2y + y^3x -xy$ for stationary points? A link to an educational resource that goes over this would be very helpful as well, as I don't even know what ...
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1answer
19 views

Are there two notions of flow?

I'm reading Jungnickel's Graphs, Networks and Algorithms. He defines the flow as a mapping $f:E\to \mathbb{R}_0^+$, which seems to mean the value of the flow of each edge, but in here: When he ...
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19 views

Is there a cost function for row equivalent matrices?

I am working on a minimization problem as follows: argmin$_x$ ||x-y||$_2$$^2$+$\lambda$||$\Psi$x||$_1$ where x and y are 2D or 3D complex arrays ||$\cdot$||$_1$ and ||$\cdot$||$_2$ are the L1 and L2 ...
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2answers
29 views

Can calculus optimization problems be turned into linear programming problems?

I found a Linear Programming textbook somewhere, and I skimmed through the first few pages. While I am not nearly enough ready to go through it, the things it dealt with seemed very much like calculus ...
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Is there Lipschitz property for subdifferential?

I'm trying to bound the quantity $\langle \nabla \Psi(x),\bar{x}-x \rangle$ above, with the bound depending on $\|x-\bar{x}\|$ and perhaps also of $\|x-y\|$ for fixed (but not varying) points $y$. ...
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2answers
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Maximizing $\log(|A|)-\text{Tr}(AB)$ for pd and symmetric $A$ and $B$

Let $A$ and $B$ be two symmetric and positive definite matrices of the same size. Then the function $$ f(A)\equiv\log(\det(A))-\text{Tr}(AB) $$ is maximized uniquely by $A=B^{-1}$. This is ...
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Preconditioning of optimization problems

This question suggests that you can precondition an optimization problem by a simple multiplicative scaling of the variables in the objective function. However, when I look up literature on ...
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How to work with difference-of-elements penalty in optimization

I am trying to solve the optimization problem $$\min_{H,S>0} \|W(H+S)-X\|^2_F+Q(H)+\eta\|S G\|_F^2$$ where $X\in\mathbf{R}_+^{m\times T}$, $W\in\mathbf{R}_+^{m\times k}$, ...
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18 views

Finding the Dual of a primal LP

Suppose that we have the following primal LP: $\min z=c^Tu+d^Tv \\ \mbox{s.t.}\ \ \ \ \ \ \ \ \ u+Av=b, u\geq 0, v\geq 0$ I want to find the dual problem of this LP but I am slightly confused as ...
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25 views

How to get the updating rules? after derivation

In the picture i brushed yellow, it dose make no sense to me to get formulas(2) and (3). If anyone could point out or give some references? Thanks a lot!
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63 views

Efficiency of a max-min problem for $\sum_{j=1}^m |b_j-a_j|$ with $a_i$, $b_j$ restricted to convex sets

Consider the following optimization problem: $$\max_{\{a=(a_1,a_2,\ldots,a_m)\in A\}}\min_{\{b:=(b_1,\ldots,b_m)\in B\}} \sum_{j=1}^m |b_j-a_j|.$$ Is computing the optimal value of this problem ...
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46 views

Finding L^1 centers of sets of probability distributions

Let $\mathcal{P}^n = \{ x \in \mathbb{R}^n : x \geq 0, \sum x = 1\}$. Suppose I have $p_1, \ldots, p_m \in \mathcal{P}^n$. I want to find an $L^1$ center for these points. i.e. $q \in \mathcal{P}^n$ ...
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41 views

applied optimization problem- triangle fence

A farmer is trying to fence off a field on the edge of a river. He has two 1km long sections of fence to use to make a triangular field. The edge by the river does not need fencing, and the fence ...
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15 views

Optimization problem-distance question

Jessica needs to get to a boat. the boat is 100m offshore. She is currently running along the beach 1km away from the closest point on the beach to the boat. She can run at 3m/s and swim at 1m/s. She ...