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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What minimizes the Chebyshev Distance?

For an arbitrary number of dimensions, I know that the mean minimizes the distance using the L2 norm and that the geometric median minimizes the distance function using the L1 norm (though I have yet ...
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181 views

Minimizing $L_\infty$ norm using gradient descent?

Curve fitting problems are solved by minimizing a cost/error function with respect to the model's parameters. Gradient descent and Newton's method are among many algorithms commonly used to minimize ...
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52 views

$AB$ is a chord of a circle $C$. Let there be another point $P$ on the circumference of the circle, optimize $PA.PB$ and $PA+PB$

$AB$ is a chord of a circle $C$. (a) Find a point $P$ on the circumference of $C$ such that $PA.PB$ is the maximum. (b) Find a point $P$ on the circumference of $C$ which maximizes $PA+PB$. My ...
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154 views

Free software or algorithm for Second-Order Cone Program

I need to solve the following optimization problem: $$ \mathbf{x}^\ast = \operatorname{argmin}_{\mathbf{x}} \Vert \mathbf{Rx} \Vert_2^2 \;\;\; \mathrm{s.t.} \;\;\; \mathbf{s}^\mathrm{H} \mathbf{x} = ...
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2answers
257 views

Value minimizing mean absolute percentage error

What value for $c$ would minimize the formula: $$\frac{1}{n}\;\sum^{n}_{i=1}\left | \frac{y_i-c}{y_i}\right|$$ given the values $y_1, ..., y_n$. For example in the mean squared error we have the ...
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65 views

computing dual LP in graph matchings

I'm having a trouble converting the following LP to a dual LP. Help on some starting steps would be greatly appreciated!
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93 views

extreme value of increasing or decreasing function

From the three problems: one, two and three, it seems that if $f(x)$ is decreasing function with $a+b+c=abc$ or $a + b + c + a b + b c + c a = 1 + a b c$ then the maximum value of $f(a)+f(b)+f(c)$ ...
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169 views

Local extrema of continous, non-differentiable function

Ok, so looking through some questions I've found this answer: http://math.stackexchange.com/a/1667/102636 containing proof, that there are either inifinitely or no local extrema of continous and ...
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65 views

Guessing Game Stochastic Optimization

This is part of another post I did, but I think it has interest in its own right: Let $Y =\{X_{1},X_{2}...X_{N}\}$ be a set of $N$ random quantities with assocated set of distributions ...
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340 views

Determine all the extrema of a function subject to a non-linear constraint.

QUESTION Determine all extrema of the function $$f(x,y) = x+ 2y $$ subject to $$x^2 + y^2 - 80 = 0$$ ATTEMPT I don't think I understand what I'm supposed to do. This was in a test and I ended up ...
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154 views

Simultaneous Maximization and Minimization

I have a function with two variables say $$g(x,y)=f(x)−h(x,y)\ $$ where $$ f(x)= ax-bx^2\ $$ and $$h(x,y)=(x+y)^2\ $$ and $$ y>=0, x+y>=0\ $$My purpose is to maximize g(x,y) for x, ...
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451 views

Using gradient descent and Newton's method combined

I have this function $f(\mathrm{X})$ where $\mathrm{X=A+B+C}$ where $\mathrm{A}$ is a diagonal element with variable $a$ on its diagonal. $\mathrm{B}$ is another diagonal matrix with variable $b$ on ...
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28 views

Minimization values for a function [duplicate]

I have got function - non linear(I thk), and a set of variable S=[(x1,y1),(x2,y2)...]. The objective is to find the value for ...
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67 views

A minimization problem

Define $$L(w,u)=\frac{1}{2}\|w-u\|^2+\beta \left\|\frac{w}{x}\right\|,~w,u\in \Bbb{R}^n$$ where $$\frac{w}{x}=\left(\frac{w_1}{x_1},\ldots, \frac{w_n}{x_n}\right)$$ $$\|x\|=\sqrt{x_1^2+\cdots+x_n^2}$$ ...
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392 views

Minima of symmetric functions given a constraint

If $f(x,y,z,\ldots)$ is symmetric in all variables, (i.e $f$ remains the same after interchanging any two variables), and we want to find the extrema of $f$ given a symmetric constraint ...
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224 views

the implication of zero mixed partial derivatives for multivariate function's minimization

Suppose $f(\textbf x)=f(x_1,x_2) $ has mixed partial derivatives $f''_{12}=f''_{21}=0$, so can I say: there exist $f_1(x_1)$ and $f_2(x_2)$ such that $\min_{\textbf x} f(\textbf x)\equiv ...
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2k views

Maximize area of a triangle with fixed perimeter

If perimeter of a triangle is $2d$, what is the length of sides so the triangle has maximal area? I found some solution using circle and angles, but I think I have to use derivatives. I need ...
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132 views

Determine the points where $f$ is has a local minimum/maximum. Multivariable calculus question.

This is not homework, but it is in my book and I find it hard to solve: Determine the points where $f$ is has a local minimum/maximum. Determine if it strong/weak and absolute/relative and ...
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215 views

Are these convex optimization problems equivalent?

Consider the optimization problem $$ \mathcal{P}_1: \qquad \min_{x \in \mathbb{R}^n} c^\top x \quad \text{sub. to } \ g(x,y_i) \leq 0 \ \ \forall i = 1,2,...,M$$ where $c \in \mathbb{R}^n$, and ...
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331 views

Showing a dual LP solves a primal LP

I originally asked this question: Does solving the LP dual SOLVE the primal LP? It was answered using an example of how the primal and dual solve each other (because of knowledge from strong ...
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3answers
756 views

A constrained linear least Frobenius norm problem:$\min_{X} \|A-XB\|_F$ subject to $Xv=0$?

Assume we are given two matrices $A, B \in \mathbb R^{n \times m}$ and a vector $v \in \mathbb R^n$. $\|\cdot\|_F$ is the Frobenius norm of a matrix. How can we solve $$\min_{X \in \mathbb R^{n ...
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569 views

How would you solve a Tikhonov Regularized Least Squares problem with nonnegative constraints?

For a Tikhonov Regularized Least Squares problem with nonnegative constraints, what are some methods that solve it? Are methods solving a Least Squares problem with nonnegative constraints and the ...
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79 views

Is $ \sum_{1 \le k \le n} (y_k - a x_k^b + c x_k^d + e)^2 $ convex?

Over at How many points to find a polynomial? it was suggested to minimize $$ f(a,b,c,d,e) = \sum_{1 \le k \le n} (y_k - a x_k^b + c x_k^d + e)^2 .$$ However I don't know if it is possible to find ...
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6k views

Max distance between a line and a parabola

I am suppose to use calculus to find the max vertical distance between the line $y = x + 2$ and the parabola $y = x^2$ on the interval $x$ greater then or equal to $-1$ and less then or equal to $2$. ...
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525 views

Understanding how to state the Karush-Kuhn-Tucker Conditions for a given problem

I'm trying to understand an example given by Nocedal & Wright (1999), pg 329, Example 12.4. According to a definition given earlier in this book: At a feasible point x, the inequality ...
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491 views

Bilinear Optimization Problem

How could I solve the following optimization problem using MATLAB or an other way? Given ${E}^{1}, {A}^{21}, {A}^{22}, {C}^{1}, {A}^{12}$ $ \underset{{C}^{2}, {E}^{2}}{min} {\left \| {C}^{2}{E}^{1} ...
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85 views

Minimial Tiling Problem on a sphere

This question is a revision of the math exchange post found here. Consider the following: A sphere, $S$, with radius $r_1$. N regions projected onto $S$, whose projections, $\left\lbrace E_i ...
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624 views

Sign restriction on the Lagrange multiplier? Why?

Say we are given a linear program where the goal is to minimize $c^Tx$ with the constraints $Ax\ge b$. Why is there a sign restriction on the Lagrange multiplier associated with the active constraints ...
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Set boundary for least-square calculation

When calculating least-squares we use the form $$Xw=y$$ This gives us an approximation, and it can happen that the resulting $w$ will be such that $Xw \lt y$ (entry-wise). Is there a way how to ...
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70 views

Upper bound of an expressions with many variables

Assume $0 < p_1 \le p_2 \le \dots \le p_{2k}$. I am looking for a (preferably tight) upper bound for the following expression: $$ \frac ...
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325 views

Underlying assumption in a Primal/Dual table

I just read in one of the questions answered by @MikeSpivey that the following table is provided in Sierksma's Linear and Integer Programming: Theory and Practice, Volume 1, page 144. ...
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337 views

Sudoku mathematically, MILP?

My homework contains a word (freely-translated) "target-function" that I should generate somehow for 9x9 sudoku solver with some MILP problem. But I am bit lost what they mean. I have sofar described ...
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817 views

Optimizing with Absolute Value Objective Function

max : $w = |q^T y|$ subject to $A y \leq b$ $y \geq 0$ Please describe how one could solve the non-linear programming prob. above by using linear programming methods. I tried changing $y$ to $y' ...
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249 views

Questions about weak duality theorem

Following are some corollaries regarding the weak duality theorem. Consider a constrained problem, $\min_{x \in X} f(x),$ subject to $g(x) \leq 0$ and $h(x) =0$. Its dual problem is $\sup_{u \geq ...
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120 views

Minimum for this function

I thought of writing this question Minimum for this function in a different way, if it helps. I want to minimize $$\sum_{i=1}^n a_ix_i + \nu \sum_{i=1}^n b_i 2^{x_i} ,$$ where $a_i \in [0,1]$, ...
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158 views

Find the range of $x$, given $y_{min} \leq y(x) \leq y_{max}$, where $y(x) $ can be any function ( Updated)

I have a series of inequalities: $$y_{1min} \leq y_{1}(x) \leq y_{1max}$$ $$y_{2min} \leq y_{2}(x) \leq y_{2max}$$ $$..$$ $$y_{nmin} \leq y_{n}(x) \leq y_{nmax}$$ Note that $x\in\mathbb{R}$ The ...
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The solution of $\min_x \max_{1\leq r \leq N} \left|\frac{\sin rMx}{\sin rx}\right|$

I am looking for the solution of $$\min_x \max_{1\leq r \leq N} \left|\frac{\sin rMx}{\sin rx}\right|$$ where $M < N$ are integers and $x \in \mathbb{R}^+$. For $M = 4, N = 6$, $f_{r,M}(x) ...
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Notion of outer normal cone and supporting cone if $x \in$ relint($C$)

In my lecture we defined the outer normal cone $ N_c(x^*)= \{ c\ \in \mathbb{R^n} : \max\limits_{x \in C} \ \ c^Tx = c^Tx^* \}$ and the supporting cone $S_C(x^*)= \bigcap\limits_{c \in N_c(x^*)} ...
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41 views

Minimization involving equality constraints

I am trying to find closed form solution to following problem \begin{equation} \begin{array}{c} \underset{\mathbf{x},\mathbf{y}}{\text{minimize}} \hspace{4mm} \big(\left( \mathbf{y}^T ...
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Definition issue with limiting directions

In Nocedal/Wright's Numerical Optimization (1999) in section 12.3 the notion of feasible sequences and related limiting directions are introduced as a starting point for the proof of the ...
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The functional take its maximal value for $y(t)=-t$

I want to show that the functional $J(y)=\int_0^1 [y'(t) \sin{(\pi y(t))-(t+y(t))^2}]dt$ ,where $y$ is a continuously differentiable function on $[0,1]$, takes its maximal value $\frac{2}{\pi}$ for ...
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How can we maximize the following functional?

$\max_{} \; \int_0^1 \left( -\frac{1}{2} \left( \lambda_1(1-t) - \int_t^1 \lambda_2(s) ds \right)^2 - 1.25 \lambda_2(t) \right)dt + \lambda_1$ s.t $\lambda_1\geq0$, and $\lambda_2(t) \geq 0$ for ...
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Do three valued basis vector elements lead to the fastest discrete Fourier transforms?

When sin() and cos() are approximated to 1, 0 and -1 in the basis vectors in a real or discrete Fourier transform the basis vectors have a lot of elements of zero or in common leading to an algorithm ...
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Difference between mean-variance and worst-case optimization for normal distribution

I have two optimization problems. 1-) Mean-variance optimization $J_{MV} = J_M - \gamma J_V$, where $J_M$ is mean, $J_V$ is the variance term and $\gamma$ is the weight on variance term. 2-) The ...
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How to determine the gradient of this cost? [duplicate]

I have asked a similar question before, but I guess I haven't provided clear information. The cost of my function $f:\mathbb{R}^5\rightarrow\mathbb{R}$ is $$f(\vec{\alpha}) = ...
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Problem about length and width of a running facility

Jacaranda Secondary College is planning to develop a $400$ metre running track facility in an unused area of the college. The rectangular site available is $100$ metres wide and $180$ metres long. The ...
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163 views

Proof that nuclear norm is convex.

For an $m \times n$ matrix, $A$, the nuclear norm of $A$ is defined as $\sum_{i}\sigma_{i}(A)$ where $\sigma_{i}(A)$ is the $i^{th}$ singular value of $A$. I've read that the nuclear norm is convex ...
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Find $\min x^TAy+b^Tx+c^Ty$ subject to $1^Tx=1^Ty=1,x\ge 0,y\ge 0$

The problem seems to be easy but I can't find a solution :( Problem: Given $A\in\mathbb{R}^{m\times n}, A\ge 0, b\in\mathbb{R}^{m}, c\in\mathbb{R}^{n}$. Minimize $f(x,y) = x^TAy+b^Tx+c^Ty$ subject to ...
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138 views

An Interesting Resource Allocation Problem

Here is the problem: \begin{array}{ll} \text{minimize} & \sum_{i=1}^N \frac{1}{1 + \textrm{exp}(C_i + x_i)}\\ \text{subject to} & \sum_{i=1}^N x_i \le R \\ & x_i \ge 0, ~ i = 1,2,...,N ...
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39 views

Looking for Method to evaluate the optimal node rate vs number of simulation rate in a Monte Carlo simulation

I am currently working on evaluating an American Option using a Monte Carlo simulation, and I am getting answers but they vary quite a bit. The two variables that I can alter are number of simulations ...