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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Find the smallest possible value of $a^4+b^4+c^4-136abc$

Let $a$, $b$, and $c$ be real numbers such that $a+b+c=-68$ and $ab+bc+ca=1156$. The smallest possible value of $a^4+b^4+c^4-136abc$ is $k$. Find the remainder when $k$ is divided by $1000$. I ...
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35 views

Weakly lower semicontinuous functional on a bounded closed and convex set

Let $J$ be a sequentially weakly lower semicontinuous functional on $C$ with values on the real line. Moreover let $C$ be a bounded, closed and convex subset of a Hilbert space $H$. Is it true that ...
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40 views

Using Lagrange multipliers to identify the Extremes of function $f(x, y)=x-y$, under condition $g(x,y)=x^2 + y^2 - 4=0$

I'm studying in preparation for a Mathematical Analysis II examination and I'm solving past exam exercises. If it's any indicator of difficulty, the exercise is Exercise 3 of 4, part $b$ and graded ...
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14 views

Existence of non-negative solution to a diagonally dominant tridiagonal system

Let $D \in \mathbb{R}^{n \times n}$. having only non-negative entries, strictly diagonally dominant (both row-wise and column-wise), tridiagonal. Show that $$\exists\; x \in \mathbb{R}^n \quad ...
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30 views

Maximise volume given inequality constraint on its dimensions without using Lagrange, KKT or Linear Programming

The problem (from Calculus for Business, Economics, Life Sciences and Social Sciences 12e): I found this and that, but they use Lagrange/KKT. What I tried: Girth $= 2w + 2h$ Maximise ...
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17 views

Maximal distance of a segment

Let a path enclosed by lines as illustrated in this figure Fig. knowing that the widths of the two paths are $\ell$ and $\ell^{\prime}$ respectively. What the maximum distance $x^{\star}$ to be able ...
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19 views

KKT conditions for nonlinear problem

I need to state the KKT conditions for the following problem: Minimise $x_1^2 + 2x_2^2$ subject to $(x_1-1)^2 + x_2^2 \le 1$ and $x_2 = 1$. I have that these conditions are: $f(x^*) \le 0$ ...
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16 views

Can lagrange multiplier(Kuhn tucker multipliers?) change in corner solution?

If we want to maximize $f(x)$ subject to two constraints, one which says that $x< c$ $c>0$, and another that says that $x\geq 0 $. Assume there are no problems with either $x=0, x>0$ or $\mu ...
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Find min $ax+by+cz$ subject to $0 \le y \le 1, 0\le z \le 1$ and $\max(0,y+z-1) \le x \le \min(y,z)$

I am seeking an elegant way to solve the following problem. Let $a,b,c$ be constant real numbers. Find min $ax+by+cz$ subject to $0 \le y \le 1, 0\le z \le 1$ and $\max(0,y+z-1) \le x \le \min(y,z)$. ...
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Feasible set and level sets

Consider the following problem: Minimise $x_1^2 + 2x_2^2$ subject to $(x_1)^2 + x_2^2 \le 1$ and $x_2 = 1$. Sketch the feasible set and the level sets of the objective function, and determine an ...
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27 views

Optimization on a grid

I worked a lot on defining the problem so I will be grateful to get input if i'm not clear enouth and I will fix the question. We have a grid made out of uniform points on $[x,y],$ $x,y\in[0,1],$ ...
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48 views

what is the maximum value of $x(x+y)^3$ given that $x^2+y^2/d=1$?

Without losing generality, we can assume $x,y\geq 0$ and then use $x$ to replace $y$. This is complicated. Instead I use $x=\sin\theta$, $y=\sqrt{d}\cos\theta$, and then I only need to get the ...
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25 views

Solving SVM classifier with two weight vectors

I am trying to implement a paper that basically proposes the following way to train two classifiers on some data with two types of labels. I do not know how to tweak existing solvers for SVM to do the ...
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15 views

Design a circuit for a function

I am so confused on this problem. We are given a function $f$ and told to design a circuit that has four inputs labeled $b_3,...,b_0$, and an output $f$, where $f = 1$ if the 4-bit input pattern is a ...
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74 views

Multivariable optimization for time to build a ship in a game, and maybe some possible application in “everyday” life

I precise first that english is not my monther tongue and I may will not be as clear as I would like, just ask me question if you need, thank you. I am playing a game (Galaxy Empire) for a while, ...
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14 views

Envelope theorem for Conditional value at risk

Let $X$ be a Gaussian random variable and suppose $f(p,X)$ is a strictly increasing and continuous function in $p \in \mathbb R$. Conditional value at risk is defined in the following way ...
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38 views

MInimum value of the sum of three numbers

if product of three numbers is 1, how do you find the minimum value of the sum of those three numbers? i tried to find the possible values of the numbers that would give a product of one but I'm not ...
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20 views

Minimising logistic loss function to find optimal matrix

Please take a look at this paper on classifying triples (re link prediction): http://arxiv.org/pdf/1510.04935v2.pdf The question is about how to solve equation 2 using stochastic gradient descent. It ...
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28 views

Optimising volume of a truncated cone

Given a slant height h and radius r1 how can I find a truncated cone with largest volume?. Is there any calculus involved in ...
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21 views

Intuition behind eigen values in Optimization Problems

The question may seem very simple, I am not able to understand the intuition behind the solution of following problem min $A\vec{v}$ where $A$ is some matrix The solution is the eigen vector ...
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13 views

Best way to get combination of elements to fill given area

I am application developer and I came across interesting mathematical problem. Let's assume we are given: dimensions of space we would like to fill: a = 3m; b = 4m; set of elements: Items = { {a0, ...
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On a maximization exercise in $\mathbb R^2$.

I am given a set $A = \{ (x,y) \in \mathbb R^2 \mid x^2 + y^2 \le 1 , x+y \ge 0 \}$ and a function $f(x,y) = (x-y)^2 (x+y)$ I am tasked with finding $f(A)$. So because $f$ is a continuous function ...
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Can I use Lagrange multipliers with redundant constrains?

Can I use Lagrangian multipliers with redundant constrains? For example, suppose I have the following problem: Find the maximum of $F(x,y,z)$, subject to $f(x,y,z)=0$ and $g(x,y,z)=0$. But you also ...
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Effect of marginalization on Gauss-Newton equations

Consider the problem of minimizing the cost function $f(x)=\eta(x)^TW\eta(x)$, where $\eta(x)=z-h(x)$ is an error function between the observations (measurements $z$) and their prediction $h(x)$, and ...
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26 views

Minimizing a quadratic function of 2 variables in quadratic region

Let $f$ be a real valued quadratic function of 2 real variables: $$f(x,y) = ax^2 + by^2 + cxy + dx + ey + f$$ How to minimize it? Subject to constraints: $$ 0\leq x \leq 1, \quad 0\leq y \leq 1 $$ ...
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27 views

find max of total area of solenoid

I'm having problems findind the best way to maximize the total area of my solenoid, of course given $l$ as the costant value of the total lenght of my wire. I can't find it on the internet, so I'm ...
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29 views

Can't solve Pontryagin's minimum principle

I try to solve a optimizing problem with the help of the Pontryagin's minimum (maximum) principle, but I must understand something wrong, can someone help me? Here is the problem: I have a moving ...
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34 views

Minimisation problem of two equations with three variables

I have 2 equations with 3 variables. I need to find the minimum points for the 3 variables. The question as below: Equation 1: $$ 0.09518x - 0.06118y - 0.06132z = 4.031 $$ Equation 2: $$ ...
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43 views

Find radius such that packing circles into a fixed rectangle maximises total area of circles

I want to pack equal-sized circles into a rectangle with width $w$, and height $h$. The total area of all of the circles should be maximised. the radius of each circle can vary, but is contrained; ...
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Proof One Can Convert Non-Symmetric Square Matrix Into a Symmetric Square Matrix

I've read in "Introduction to Optimization" by Chong Et. Al that a non-symmetric square matrix can always be written as a symmetric square matrix. (pp.26) How does he know this? Is there a proof for ...
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Controlling a flying vehicle with multiple thrusters

I'm working on a problem involving a vehicle with $n$ rocket engines, as seen here: The task is, given the desired force $\vec F$ and torque $\vec \tau$, calculate the optimal thrust for each ...
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Load balance N customers over K servers with different capacities

Let's say we have N customers that supply a stream of requests, but each customer i supplies different number of requests per minute - $R_i$. All requests are identical in terms of the amount of ...
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How do computer programs find roots of high-degree polynomials?

My question is motivated by curiosity about the optimization of high-degree polynomial functions. Let's say your experiment data are modeled by a non-trivial 15th degree polynomial. Taking the ...
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21 views

l1 regularized minimization with equality constraint in ADMM

In section 6.3 of this note there is a method for minimizing a loss function with l1 regularization. i.e. minimize $l(\bf{x})+\lambda||x||_1$ How can I add the equality constraint $\sum\limits_{i} ...
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Obtaining zeroes in a matrix

I have a large matrix , with elements somewhere between 0 and 0.4 . I want to apply local unitaries, that is, unitary matrices of the form: $$U_{L} = U \otimes U \otimes U$$ in order to make my ...
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Scaled proximal operator for proximal Newton method

The scaled proximal operator was introduced as an extension of the (regular) proximal operator: $prox^H_h(x) = \arg\min_y h(y) + \frac{1}{2}\|y-x\|^2_H$. (See ...
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Visualising the dual function

Consider the linear programming problem: min $-x_1-2x_2$ s.t. $3x_1+2x_2-6\leq 0$ $ -x_1+2x_2 -4\leq 0$ $0 \leq x_1 \leq 3/2, 0 \leq x_2 \leq 3/2$ Find the Lagrange dual objective function ...
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How to find the maximum of this function of multiple variables?

Let $\mathbf{x}$ be a vector in $[a,b]^n\subseteq \mathbb{R}_+^n$ and $\mathbf{A}=[a_{ij}]$ a matrix in $\mathbb{R}^{k\times n}$. Let the rows of $\mathbf{A}$ be denoted by $\mathbf{A}_{i}$ for all ...
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How to find the number of possible solutions of LP problems?

Let us assume that we have a linear optimization problem (LP) that has multiple optimal solutions. I would like to know if there is a solver or an algorithm that can provide the number of optimal ...
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Dynamic “Assignment Problem” (Hungarian Algorithm Extension?)

TL;DR: Trying to optimize assignments using Hungarian algorithm, but cannot determine costs until all assignments have been made due to dependencies. Using the terminology from Wikipedia's Assignment ...
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How to “separate” a matrix into two vectors?

I have a matrix $M$ and I would like to find two vectors $u$ and $v$, that minimize $$ \sum_{i,j} (M_{i,j}-u_iv_j)^2 $$ How can I do this (numerically)? Actually this is very simplified ...
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Maxima and Minima: $y = x^4$

Given a stationary point, I was taught to test if it was a maximum or a minimum using the concavity test, i.e. If $f''(x)>0$: concave up (thus a local minimum) If $f''(x)<0$: concave down ...
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Solution of quadratic optimization with linear constraints

Hi, I want to solve a quadratic optimization problem defined as $\min \|\bf{a^T}\bf{x}\|^2_2$ $s.t. \|\bf{x}\|_1=1$ $\ \ \ \ \ \ \ \ x_i\ge0,\ i=0,1,...,n-1$ $\ \ \ \ \ \ \ \ \bf{b}^T\bf{x}-C\le0$ ...
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1answer
33 views

Is my solution to this basic calculus optimization problem correct?

I needed assistance checking a solution to a calculus problem. Consider the graph of the function $f: [-\frac{\pi}{2}, \frac{\pi}{2}] \rightarrow \mathbb{R}$ given by $f(x) = \cos(x)$. Note that it ...
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Maximum of the product of two poisson mass functions

I have two questions regarding maximising the following function defined for $x, y \geq 0$: $f(x, y) = \displaystyle \sum_{i = 0}^{\infty} \frac{x^i e^{-x}}{i!} \frac{y^i e^{-y}}{i!}$ when $x, y > ...
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Is it possible to use regularization to minimize the (expected) number of non-zero digits in a number?

This question may be slightly related to this question on length of the representation of a number in a certain basis. Introduction / Background In image and video coding, particularly the ...
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constrained heat equation

Consider the following constrained minimum energy problem for 1-D heat equation for $x\in[0,1],t\in[0,\infty)$: $$u(x,t)=\underset{{0\leq u(x,t)\leq1}}{argmin}~\frac{\partial}{\partial ...
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Why is $f = f_{0} + \sum_{i}\alpha_{i}X_{i} + \frac{1}{2}\sum_{i}^{n} \sum_{j}^{n}A_{ij}X_{i}X_{j}$ the standard quadratic form in n dimensions?

The claim that $$f = f_{0} + \sum_{i}\alpha_{i}X_{i} + \frac{1}{2}\sum_{i}^{n} \sum_{j}^{n}A_{ij}X_{i}X_{j}$$ is the standard quadratic form for $n$ dimensions, where $\alpha$ is some $ 1 \times n$ ...
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Maximum area of a rectangle whose vertices lie on ellipse $x^2+4y^2=1$

Maximum area of a rectangle whose vertices lie on ellipse $x^2+4y^2=1$. I try to do it by lagrange multiplier as $F(x,y,t)= xy + t(x^2+4y^2-1=0)=0$. Differentiating w.r.t to x,y and solving i get ...
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KKT conditions for a convex optimization (optimal crowdsourcing with budget constraint)

I am having some troubles deriving the optimal solution of the following convex optimization problem, $w_j$, $c_{ij}$, and $B$ are fixed and non negative. \begin{align} & ...