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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LP: Add extra costs in the objective function for every variable which is not equal to 0

I am trying to optimise an LP problem but extra costs should be added if a variable is larger than 0. For example if we have the following objective function: $min \text{ } 2X_1 + 3X_2 + 3X_3 $ ...
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2answers
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If $f(x)=\frac{1}{\pi}\left(\arcsin x+\arccos x+\arctan x\right)+\frac{x+1}{x^2+2x+10}\;,$ Then $\max$ value of $f(x)$

If $\displaystyle f(x)=\frac{1}{\pi}\left(\arcsin x+\arccos x+\arctan x\right)+\frac{x+1}{x^2+2x+10}\;,$ Then $\max$ value of $f(x)$ $\bf{My\; Try::}$ Here Domain of $\arcsin x\;,\arccos x$ is ...
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Kullback-Leibner divergence true distribution

I have an image with an object which I treat as 2-dimensional Gaussian random vector with mean equal to the center of the object surrounded by, roughly, 3-sigma ellipsoid. On the other hand I feed the ...
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6 views

Literature on Geometric median (or Fertmat-Webber problem)

For a finite set $X\subset \mathbb R^n$ the geometric median is defined as the point in $\mathbb R^n$ for which the sum of distances to all points of $X$ attains its minimum. Here is a wiki article: ...
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How do I minimize this quadratic program with these constraints? [on hold]

My objective function that I am trying to minimize is \begin{align} \underset{\mathbf{A},\mathbf{b}}{\text{min}}\hspace{4mm} (\mathbf{Ab})^T \mathbf{Q}(\mathbf{Ab}) \hspace{4mm} s.t ...
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1answer
16 views

Intuitive understanding of Maximin principle

From the the book page $324$, does someone could explain to me the Theorem $2$. Maximin principle? I have a bit of difficulties to well understand how works this theorem. A simple example would be ...
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Properties of unit vector scaling

What properties are kept when we scale a vector to unit length, i.e. $\frac{\mathbf{v}}{||\mathbf{v}||_1}$? Imagine that we have an unconstrained optimization problem, and we obtain as solution $x_i ...
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1answer
91 views

What is the maximum of $x^3y^3 + x^3z^3 + y^3z^3$ subject to $x+y+z=1$?

All variables are positive reals. This is a math competition problem. I've tried solving it using boundary value optimization, but it's not elegant at all. Thanks for any ideas.
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1answer
24 views

Find the absolute max and min of a multivariable function on a bounded by a circle?

So i do understand everything up the square rectangle, in the photo here i mean, how did he come up with $(±2,0), (0,±1)$ is it because of $g(2cos x, sin x)$ and if that is the case why would he ...
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2answers
54 views

Shortest distance as measured in norm $||\cdot ||$ from point to a sphere in norm $||*||$

I recently found this theorem, which is used in some clustering algorithms: Let $x,v \in \mathbb{R}^p$, $r>0$, $||\cdot ||_{\ast}$ be a given norm on $\mathbb{R}^p$ and $\partial ...
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2answers
37 views

Minimizing an integral — Hilbert space

Find the real values of $a, b$ which minimize $$\int_1^{\infty} \left| \frac{1}{x^2} - a \frac{1}{x^3} - b\frac{1}{x^4}\right|^2 \; dx.$$ Hint : Work in an appropriate Hilbert space. Here is ...
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1answer
23 views

How to solve this optimization question with the Extreme Value Theorem?

Consider the region in the x-y plane that is bounded by the x-axis and the function $f(x)=b-ax^2$. Construct a rectangle whose base lies on the x-axis and is centered at the origin, and whose sides ...
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15 views

How can I show the corresponding dual solution is unique when the given primal solution is nondegenerate, basic feasible?

the given problem is to show that if $x_1,...,x_n$ is a nondegenerate basic feasible solution of the primal LP max $\sum_{j=1}^{n}c_jx_j$ s.t. $\sum_{j=1}^na_{ij}x_j\leq b_i, \forall ...
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1answer
16 views

Rectangular prism optimization using extreme values

A box with a rectangular base, whose length is twice its width, is to have a closed top. The area of the material in the box is to be $192in^2$. What should the dimensions of the box be in order to ...
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2answers
27 views

Binary optimization problem

I am facing the following problem: Let P be a fixed m x n finite matrix and D be a matrix of ones and zeros with the same dimensionality as P plus the following constraints: sum of row entries ...
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20 views

Nonlinear optimization with constraints; is changing variables an reliable approach?

I have a optimization problem as follows, $$ \begin{array}{cll} [\hat{x_1},\hat{x_2},\hat{x_3}] = & \text{argmin}_{x_1,x_2,x_3} \sum_{i = 1}^N \sum_{t = 1}^T \left[ \ln(f_{i,t}(x_1,x_2,x_3)) + ...
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+150

Weight Modification for Computationally-Efficient Nonlinear Least Squares Optimization

There was a time where I could figure this out for myself, but my math skills are rustier than I thought, so I have to humbly beg for help. Thank you in advance. I am solving a weighted nonlinear ...
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22 views

Comparing the task complexity of installing three different offenses for American style football in three days

I want to identify the inherent difficulty of installing three separate American rules football offenses by their complexity of practice schedules in three days then relate those offenses back to one ...
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1answer
26 views

Exponent to maximize the expression $log_b \left(a\frac{b-1}{b^k-1}\right)$

Given $ a, b \in \mathbb N $, how to maximize the expression $$ log_b \left(a\frac{b-1}{b^k-1}\right) \in \mathbb N $$ Put differently, what is the minimum $k \in \mathbb N $ verifying $$ ...
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297 views

How did the answer key get $h=40-2r$?

A cone has radius of $20\ \rm cm$ and a height of $40\ \rm cm$. A cylinder fits inside the cone, as shown below. What must the radius of the cylinder be to give the cylinder the ...
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1answer
56 views

Engineering/mathmatics question

I have an equation $M(x)= -15.328x^2+176.44x-352.88$ (a parabola) and also $V(x) = -30.657x + 176.44$. I want to know how to find $x$ where the values of $M$ and $V$ combined are the lowest, I'm ...
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28 views

Find the absolute max and min values of a multivariable function bounded by a circular boundary

Find the absolute minimum and maximum values of $f (x, y) = xy e^{−2x^2 −2y^2}$ on the set $\Delta = $ {$(x,y)\in\mathbb{R^2} | x^2+y^2\le1$} i know i should take the partial derivatives and set ...
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Can Extragradient method be expressed with proximal steps?

As we know, for solving saddle point problems, forward-backward algorithm is generally not guaranteed to converge. But extragradient method converge Structured Prediction via the Extragradient Method ...
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2answers
23 views

What is the name of this (possibly classical) combinatorial optimization problem?

I have a finite number of sets $S_i$, each of the sets costing $p_i$ and containing some elements. Given the budget $b$ I want to select number of those sets to maximize $|S_{k_1} \cup S_{k_2} \dots|$ ...
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22 views

Global maximum for constrained optimization of concave function

Suppose I maximize function $x_{1}-f(x_{2})$ where $f$ is strictly convex so $\frac{df}{dx}>0, \frac{d^2f}{d^2x}>0$. Also here is a set of linear constraints in a form $g(x_{1},x_{2})\leq0, ...
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3answers
32 views

Maximize system of linear equations

Suppose you have the system $$ \begin{bmatrix} 4 & 3\\ 1 & 7\\ 5 & 9\\ 2 & 4\\ \end{bmatrix} \begin{bmatrix} x\\y\end{bmatrix}=\begin{bmatrix}b_1\\b_2\\b_3\\b_4\end{bmatrix} $$ How ...
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1answer
29 views

Why are most Lagrange multipliers zero in the SVM solution?

I read everywhere that a non-zero Lagrange multiplier $\lambda_i$ signifies that the corresponding point $x_i$ is a support vector, but I can't see how a support vector and a non-support vector have a ...
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Numerical methods and KKT in NLP

I am studying numerical methods and NLP. I started with gradient based methods, newton methods and KKT conditions. I found the following sentence: A local minimum is found by solving KKT conditions, ...
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Trying to sell the most batches of animals using linear programming

I'm trying to sell the most batches of animals... Let's say I have 200 dogs, 100 cats, and 100 ferrets. ...
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+50

Problem with finding Karush-Kuhn-Tucker points and checking for global or local minima.

I need to solve the following optimization problem $$\begin{align*} & \mathrm{Min}:\quad f(x_1,x_2)=x_1-10x_2\\ & \mathrm{subject \ to}: \quad x_1^2 -x_2 \geq 0\\ & \qquad \qquad ...
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1answer
21 views

Position of vertices of right triangle inscribed on $x^2+4y^2=1$ with maximum area using Lagrange Multipliers

I am asked to find, using Lagrange multipliers, the position of the vertices of a right triangle inscribed on $x^2+4y^2=1$ that has the maximum area. The two legs of the triangle (which are not the ...
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17 views

check for relationship duplicate numbers

I 'm a programmer C# and look for one to formulate mathematical search numbers related to avoid duplicate relationships I need to make a relationship of users in the database and for that I would not ...
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9 views

Which optimization method when Hessian is singular?

I am trying to optimize a non linear function of four variables for which the Hessian matrix is always singular (pairs of columns / lines are colinear). I wanted to use a Newton method until I checked ...
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How to reduce the number of (overlapping) constraints in a linear program?

I am trying to solve a linear program with more than 7 million constraints which could not be solved on my computer (In total around 5000 variables). In the constraints there is a overlap between ...
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4answers
87 views

Find the minimum distance to move an ellipse to be inside another ellipse?

For the problem of ellipse intersection, I would like to know an accurate "general, including the cases of two non intersected ellipses, and non aligned ellipses" method to calculate the minimum ...
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3answers
81 views

Hottest and coldest points on a heated circular plate (use Lagrange multipliers)

A circular plate given by the relationship $x^2 + y^2 \leq 1$ is heated according to the spatial temperature function $T(x,y) = 2x^2 + y^2-y$. Find the hottest and coldest point on the plate using ...
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SVM optimality criterion in Bottou, Lin (2006)

My question relates to an alternative optimality criterion for an SVM dual solution derived in Bottou, Lin (2006) in pages 8 and 9. Let: $\alpha^* = (\alpha_1^*,\dots,\alpha_n^*)$ be a dual ...
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43 views

Dual of a maximization problem

We have a positive, smooth, increasing concave function $f:\mathbf{R}^n\to \mathbf{R}^+$ and $k$ smooth, increasing constraint functions $f_i:\mathbf{R}^n\to\mathbf{R}$. I've recently encountered two ...
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30 views

Given a demand function of Q=25─0.5P, suppose the MC=AC=$20. [on hold]

Given a demand function of Q=25─0.5P, suppose the MC=AC=$20. a. Find the price, quantity, and profit under monopoly? b. Find the price, quantity, and profit under perfect competition? c. Find and ...
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Optimizing Overwatch Team Composition by Player Hero Preference [closed]

I am wordy by nature - my apologies. My attempt at a TL;DR - I want to design a small tool that optimizes the team composition of a video game based on minimizing the sum of provided player ...
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How do I specify the inverse of a correlation matrix?

To specify a correlation matrix $\in \mathbf{R}^{n\times n}$. There are $n(n-1)/2$ free elements. If I wanted to specify a matrix that is the inverse of some correlation matrix, how should I specify ...
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1answer
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Optimization algorithms for Distribution and Logistics scenario

I am looking for a way to express the following logistics/distribution problem as an equation that can be run thru a solver to find an approximate solution. The problems is described as follows: ...
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25 views

Bounding the off-diagonal entries of a matrix

The Pauli matrices are $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, $X = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$, $Y = \begin{bmatrix} 0 & -i \\ i & 0 ...
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Finding the right $\sigma$-algebra. Question on uncertainty related to the secretary problem.

I'm working on a problem related to the secretary problem. Let me give a short overview on the topic I research: You are supposed to choose the best item presented to you in a row of n items. Any ...
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10 views

incremental knapsack

Is there a way to compute the knapsack problem incrementally? Any approximation algorithm? I am trying to solve the problem in the following scenario. Let D be my data set which is not ordered and ...
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1answer
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How to convert a non-linear constraint to a linear constraint for integer programming?

I have non-linear scheduling model and I want to convert it to a linear model. But I have no idea about how can I do it. The nonlinear constraint is: For each $i, i'\in I$ and $j, j' \in J$ and $q, ...
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How to linearize this optimization problem?

I have a nonlinear optimization problem. How to solve this? $\sigma_i$ and $\rho_i$ are the optimization variables. ...
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104 views

Proving that a sphere has a minimal surface to volume ratio using Calculus of Variations

I know the problem is traditionally solved via the isoperimetric inequality, but I was hoping to solve it by minimizing a surface of revolution subject to a volume constraint. The surface area of a ...
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Deriving the E-Step and M-Steps of the EM-Algorithm?

Insects of a certain species were exposed to cold temperature and how long the insects survived was recorded. The survival times of 9 of the 10 insects, in hours, are given below. 0.8, ...
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Is convexity of the objective function sufficient for a local maxima to be a global maximum?

In my problem, I have to maximize a convex function $f(x_1,x_2,\cdots,x_n)$ subject to two equality constraints $g_1=0$ and $g_2=0$. As usual, I constructed the Lagrangian ...