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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How to interpret multiple critical points (from Lagrange multipliers) that all give a maximum value

If I have 6 critical points, 3 of which give the same maximum possible value of a function f(x,y,z), subject to a constraint g=c, is there something more to say about this solution -- or we just ...
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28 views

Minimizing long equation with hyperbolic functions

In physics book that I am reading it is said that minimizing the expression $$\phi = - N T k \log (2 \cosh(H \beta)) - \frac{J N}{2} z \tanh^2(H \beta) + H N \tanh(H \beta) $$ with respect to $H$ ...
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The maximum value of PA.PB.PC

Let A,B,C be the vertices of a triangle inscribed in a unit circle, and let P be a point in the interior or on the sides of the triangle ABC. Then the maximum value of (PA)(PB)(PC) equals to? I could ...
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Solve this set of Lagrange multiplier equations,

I'm trying to solve $$(yz,xz, xy) = (\lambda\frac{2x}{a^2},\lambda\frac{2y}{b^2},\lambda\frac{2z}{c^2})$$ with the constraint equation $$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}=1$$ ...
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What is the maximum value of work done by this force field?

An object moves in the force field $F=yz\hat{i}+zx\hat{j}+xy\hat{k}$ starting at the origin and ending at some point $A(\xi,\eta,\zeta)$ that lies on the surface ...
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15 views

Optimizing space for many shapes within an irregular shape

So let's say in a state, there are 50 schools dispersed throughout, given by Latitude Longitude points. How would we create distinct zones that optimize the space around each school? The goal is to ...
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23 views

How to find the point in convex set $C$ that is closest to $y\notin C$?

How to find the point in convex set $C$ that is closest to $y\notin C$? $C=\{ x\in \mathbb{R^2}:(x_1-1)^2+(x_2-1)^2\le1 \}$ and let $y\notin C $ but $y\notin \mathbb{R^2} $.
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Intuitive way to find the minimum surface in $\mathbb R^3$?

Suppose we have two arbitrary closed curves which intersect neither each other nor themselves. By intuition, I guess that the minimum surface ending at boundaries is unique and it is achieved by this ...
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4answers
65 views

What does it actually mean if a cost function is differentiable?

I am just learning about optimization, and having trouble understanding the idea behind differentiating cost functions. I have read that for standard optimization problems, the cost function needs to ...
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Determine Critical points in optimisation problem

So I have this problem where I am supposed to calculate the max and min value of a function $f(x,y)=x+2y$ restricted by the disk $x^2+y^2\le 1 $. I have calculated the $df/dx $ and $df/dy$ and they ...
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Single nonzero value constraint formulation in linear programming problem statement

I'm trying to write a linear programming problem statement. Values of the solution vector have a bound constraint: $0 \leq x_i \leq 1$. Another constraint is that if we take a predefined subset of ...
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Optimization problem shortest path distance and critical node detection problem (interdiction).

I am trying to formulate this optimization problem, max $d_{ij}$ where $d_{ij}$ is the shortest distance between active nodes i and j. However my problem is connecting my decision variable with the ...
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51 views

Can this be expressed in terms of linear constraints?

I'm attempting to find a matrix $X$ that minimizes some function $f(X)$ subject to the constraint that $$ X=W A Z $$ where $A$ is a given non-negative matrix with rows that sum to 1, and $W$ and $Z$ ...
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32 views

Use graphical methods to solve the linear programming problem. Maximize:

Use graphical methods to solve the linear programming problem. Maximize: $z= 4x+2y$ subject to : $x-y\le 7$ $19x+12y\le 228$ $18x+18y \le 324$ $x\ge 0,y\ge 0$ the max is ?? when x= ?? and ...
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53 views

Find the minimum of a function for only positive values of the vector variable

Let variable vector $\vec{q}$ of size $m\times1$, and its diagonal counterpart $m\times m$ matrix $Q=diag(\vec{q})$, for some $m\in\mathbb{N}$. Define fixed parameter $n\times1$ vectors $\vec{p}, ...
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Relaxation of non-convex QCQP with one quadratic and one linear constraint

According to Boyd we know that a non-convex QCQP problem with one quadratic constraint has strong duality with the relaxed SDP or Lagrange counterpart. (check "Convex Optimization" by Boyd, Appendix ...
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Calculating min/max of a multivariate function on a region

This video shows an example of how to find the absolute maxima and minima of the function $f=xy+y^2$ at the region $\{(x,y):|x|\leq1,|y|\leq2\}$. I understand why he set $f_x, f_y$ to $0$, checked ...
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super-additive, sub-additive, and shapely value limitations?

I am working on the coalition formation. Most of the scientist used concept of shapely value for distributing the utility among the members of coalition. Up to my understanding, shapely value is good ...
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116 views

Maximization of sum of functions

Let $w,a\in R^n$, and $B\in R^{n\times n}_{++}$ (the set of $n\times n$ positive definite matrices). We know that the following function (which is a specific form of the Rayleigh quotient) has a ...
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28 views

A question about a proof in nonlinear programming book

I have a question about the proof of Proposition 1.2.1 (Stationarity of limit points for gradient methods) in the nonlinear programming book (2nd edition) by Bertsekas. At the beginning of the proof ...
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Max value of trignometric function $\sin \left(x+\frac\pi6\right)+\cos \left(x+\frac\pi6\right)$

Question: The maximum value of $\sin \left(x+\dfrac{\pi}{6}\right)+\cos \left(x+\dfrac{\pi}{6}\right)$ is at what value of $x$. I solved the problem by setting the slope of the function to zero and ...
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38 views

Curve fit minimizing the sum of the deviation

I'm fitting a curve taking the smaller sum of deviations for each parameter tested, the smaller sum returns me the parameter that gives the best fit. Here is the algorithm for a test $f(x, ...
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The dual function of composite functions

Given $X$ $Y$ are two finite dimensional Hilbert space. Let $K$: $X\to Y$ be linear and $F$: $Y\to \mathbb R^+$ is convex. Let us use $F^\ast$ to denote the dual (conjugate) function of $F$. Recall $$ ...
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Calculating amount of cubes that fit in a sphere

I know that the problem of finding out how many spheres can fit in a cube is a commonly asked and well documented one, but I am struggling to find anything on the inverse of the problem, namely: ...
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29 views

Sensitivity Analysis: Calculating Allowable increase/Decrease for a Binding Constraint

Let say we have the following equations: Objective Function = $7T+5C$ Contraints $3T + 4C \le 2400$ $2T + C \le 1000$ $C \le 450$ $T\le 100$ How would we calculate the allowable increase and ...
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Taking derivate wrt a vector

I'm trying to read through Wiki's description of the Levenberg-Marquardt algorithm. I've taken linear algebra, but I've always been fuzzy about taking derivatives with respect to a vector and just ...
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Distance between a plane and set of points

Suppose $m$ data points belonging to a class represented by matrix $A$. Therefore, the size of matrix $A$ is $m\times n$. In addition, suppose $w\cdot x + b=0$ be equation of a plane in ...
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51 views

Methods to Minimize Functions and Integrals over $\mathbb{N}$.

In a paper I'm writing, I have to minimize a messy function $f(\mu,n)$ where $\mu \in \mathbb{R}$ and $n \in \mathbb{N}$. That is, given $\mu \in \mathbb{R}$, I need to minimize the one variable ...
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65 views

Is this a game theory problem or optimization problem?

Consider a problem that looks for a $x$ that can make the following problem into some equilibrium state (similar to an equilibrium solution to a min-max problem in game theory) $$ \max_x f(x)$$ $$ ...
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Shaking a box of rocks (Optimal Packing)

My coworker was telling me that when he plants seeds on his farm, he puts them all in a large container on the tractor and after a period of just driving, the seeds are more densely packed than when ...
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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 ...
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Finding extrema of function of three variables

So i have to study this function and find out if there are any local or absolute extrema : $ f:\mathbb{R}^3 \rightarrow \mathbb{R} :$ $$ f(x,y,z)=2-\left(z-\sqrt{x^2+y^2}\right)^2 + ...
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If this problem is not unbounded, what's wrong in this dual derivation?

In a paper with 100 citation, Robust Support Vector Machine Training via Convex outlier Ablation, a convex relaxation is used. In this paper, a form of robust svm proposed: \begin{align} \min_{0\leq ...
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Finding limit and maximizer

Let $f(x):=x^\alpha - k \cdot (x+c)^\alpha$, defined for $x>0$, where $k,c>0$ and $0<\alpha<1$. Question: solve $\max_{x>0} f(x)$. Below are my thoughts: Calculate $f'(x) = \alpha ...
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Expected number of times a set of 10 integers (selected from 1-100) is selected before all 100 are seen

Suppose I have a set of 100 integers. I randomly choose 10 of those, make a note of which ones I selected, and repeat the process. What is the expected number of times this process must be repeated ...
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Optimal allocation in network

We want to analyse specialization matters in a given network (N,g). Nodes represent individuals that can produce goods and services (just like in our usual economy) and that can be consumers too. ...
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prove $\max \mathbf{a}^T \mathbf{b}_i \leq \mathbf{a}^T \mathbf{c}_i$

Can I prove the following (with or without assumptions, e.g. all the elements in $\mathbf{a}$ or $\mathbf{b}$ are positive? $\max \mathbf{a}^T \mathbf{b}_i \leq \mathbf{a}^T \mathbf{c}_i$ where ...
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Is it possible to prove $\arg\min_a f(\max(\mathbf{a}^T\mathbf{b}_i)) = \arg\min_a f(\mathbf{a}^T\max(\mathbf{b}_i))$

I have the following optimization problem, $ \arg\min_\mathbf{a} f(\max(\mathbf{a}^T\mathbf{b}_i))\;\; i=1, \dots ,N$ where $\mathbf{a}$ and $\mathbf{b}_i$ are vectors of dimension $d$. Let $B = ...
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Graph Theory: Find optimal subgraph that contains a certain node and a fixed number of nodes

I have a connected graph $G$ and a real-valued function $f$ on sub-graphs $G' \subseteq G$. Given a node $n \in G$ and a positive integer $s$, I am looking for the connected subgraph $G' \subseteq G$ ...
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14 views

Optimization on SE(3) with matrix logarithm

I am trying to optimize the following equation on manifold SE(3). $$Z(e^{\epsilon}) = \text{logm}{((e^{\epsilon}X)^{-1}W^{-1}e^{\epsilon}XY)}$$ Note that $W, X, Y, e^{\epsilon} \in SE(3)$ and $W, X, ...
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Linear objective with quadratic constraints

I have the problem $$ \text{maximize } f= c^Tx \\ \text{subject to } x^T Q x \leq 1 \\ x,c \in \mathbb{R}^n \text{ , } Q \in \mathbb{R}^{n \times n} $$ and $ Q $ is additionally symmetric positive ...
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Regression linearization to apply Gauss-Newton

I want to try and use Gauss-Newton in order to estimate a solution to the regression problem with normalizing factor $$\min_{x \in \mathbb{R}^n}: \|y - Ax\|_2^2 + \lambda\|x\|_1.$$ To do this, I have ...
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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} ...
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53 views

Optimize for happiness and equality

I'm trying to solve an optimization problem: There are $N$ students who can choose to enroll into $C$ courses, each of them has a set of 3 preferences $P = \{c_1, c_2, c_3\}$ about the courses they ...
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57 views

Upperbound for $\sum_{i=1}^n\frac{1}{x_i^2}$?

Suppose that $x_i>0$, $i=1,\ldots,n$. I'm looking for an upperbound (doesn't have to be particularly tight) of $\sum_{i=1}^n\frac{1}{x_i^2}$ in terms of some symmetric function of ...
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Sequential information discovery in minimum number of steps when some items have information about other items

There are N items, say three: call them A B and C. For each item, there is an associated bit (0 or 1) and there is a prior probability that the bit is 1, call them p(A), p(B) and p(C). There is some ...
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Why does minimizing $H[f] =\sum^{N}_{i=1}(y_i-f(x_i))^2+\lambda \| Pf \|^2 $ leads to solution of the form $ f(x) =\sum^N_{i=1}c_iG(x; x_i)+p(x)$?

I was reading the following paper of dimensionality reduction (1) and also one on theory of networks for approximations and learning (2) and was trying to understand how the regularization problem ...
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17 views

Integer problem to minimize cuttings

A company has to make 4 items in the given quantities. item1 =4 item 2=2 item3=1 item 4=1 Te surfaces has to be covered in plywood.The company has got 3 ...
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1answer
64 views

Folding a paper such that the size of one sides be as minimum as possible?

Suppose that we have an A4 paper like this: How to fold this paper such that the bottom-right corner overlap the left edge of the paper and that the size of AB side be as minimum as possible. It ...
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Complexity of the Dinic, Malhotra, Kumar and Maheshwari (DMKM) method

I'm asked to prove that the complexity of the DMKM method is $\mathcal{O}(m\cdot n^{\frac23})$ if all capacities in a network are equal to 1. I have no clue where to start, can anyone give me a hint? ...