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.

learn more… | top users | synonyms (5)

0
votes
0answers
26 views

Finding saddle points with some optimization

Consider a function that has multiple saddle points e.g. $$(x+y)(xy+xy^2)$$ We know for functions of two variables we can use the 2nd derivative test to locate saddle points $$Saddle ...
0
votes
1answer
15 views

KKT for not convex problems

In my optimization course we learned something about KKT for not konvex problems: $$min \; f(x)$$ $$s.t. \; c(x)=0$$ $$d(x)\geq 0$$ $$f(x): \mathbb{R}^n\rightarrow \mathbb{R}$$ $$c(x): ...
1
vote
0answers
14 views

Optimal value of decision variable leads to inconsistency

$\epsilon$ is a random variable with support in $(0.8,0.95)$ and pdf $f(\epsilon)$. The following equation arises out of a business problem: $ENP=800*A*E(\epsilon)+ 9000 - ...
0
votes
0answers
10 views

How to express these in AMPL. [closed]

max $(\prod_{t=1}^Tx_t)^{1/T}$ which is also the geometric mean of vector ${x}$, i.e., geo_mean($x$). How can I express this in AMPL?
4
votes
2answers
45 views

Minimizing the Frobenius Norm

I would like to minimize the following expression with respect to matrix X: $$\left \| A-BX \right \|_{F}$$ where A and B matrices are given and all the matrices have positive integer elements. Any ...
1
vote
0answers
25 views

Minimization using logarithmic barrier function

I'm thinking of the quadratic problem(QP) \begin{align} &\underset{x\in \mathrm{R}^n}{\mathrm{Minimize}}\ \ \ \frac{1}{2}x^\top{}Qx + f^\top{}x\\ &\mathrm{subject\ to}\ \ \ \ a_ix \leq b_i\ ...
3
votes
1answer
365 views

Maximum likelihood estimators, hypergeometric and binomial

I'm trying to solve a two part problem. The set up is as follows: consider a bag with $\theta$ red marbles and $7-\theta$ blue marbles, with $\theta$ being unknown. Let $x$ denote the number of red ...
0
votes
1answer
73 views

A binary min-max optimization problem

I encountered a very special optimization problem for a practical application. We have a variable $$\mathbf{s}=(s_1,s_2,s_3, s_4)^T$$, where $s_i$ can only take $1$ or $-1$, and we also have a ...
0
votes
0answers
24 views

Minimization problem involving a set of prime numbers and modular arithmetics

I'm a student working for curiosity on a general minimization problem where I suppose that there is no efficient algorithm for solving it. I'd like to ask for your valuable advice. Let $P$ be a set ...
0
votes
3answers
208 views

Lagrange multipliers - finding maximum/minimum

I have solved the question, and obtained the critical points, but don't know how to show its a maximum or minimum of a function. I don't understand other answers because symbols confuse me so much and ...
0
votes
1answer
800 views

How do I construct the Jacobian for use in a Levenberg-Marquardt algorithm.

I am working on a 3D reconstruction system and I am looking to use a Levenberg-marquardt algorithm to do bundle adjustment. I am not too sure about how LM works and what it requires. The model I am ...
1
vote
0answers
54 views

Prove $\lambda=\min_{i = 1,\ldots, n}\max_{0 \le k \le n-1}\left(\frac {p_i(n)-p_i(k)}{n-k}\right)$

Prove the minimum directed cycle mean cost satisfies: $\lambda = \min_{i = 1,\ldots, n} \max_{0 \le k \le n-1} \left(\frac {p_i(n) - p_i(k)} {n-k}\right)$ using the Bellman-Ford algorithm. Let ...
1
vote
1answer
27 views

Optimization using Karush-Kuhn-Tucker conditions

min $y^Tx$ subject to $\|x\|^2 \le 1$ where y is a nonzero vector in $\mathbb R^n$ I rearrange the constraints so that the RHS is $0$. New constraint: $x_1^2 + \cdots + x_n^2 - 1 = \|x\|^2 - 1 \le ...
0
votes
0answers
27 views

How does this optimization problem satisfy Karush-Kuhn-Tucker Conditions?

I am following Andrew Ng's course notes on Support Vector Machines at: http://cs229.stanford.edu/notes/cs229-notes3.pdf There is something in these notes which I do not understand. SVM's basic ...
1
vote
1answer
663 views

primal to dual solution conversion ??

i have an optimization problem $$\text{ maximize } z=3x+4y$$ $$\text{ such that: } x+y ≤ 450 \text{ and } 2x+y ≤ 600$$ the optimal solution to this problems comes to be $x=0$; $y=450$; $p=150$ ...
3
votes
2answers
51 views

Intuition about gradient

https://en.wikipedia.org/wiki/Gradient Gradient is a vector which we can obtain from any differentable function taking its partial derivatives. From Wiki: "...the gradient points in the direction of ...
1
vote
0answers
22 views

What is $F_P$ and $E(P)$?

I'm reading Handbook of Graph Theory: At this section, he speaks about $F_P$ and $E(P)$. It's not really clear what they are. I guess there is enough context for someone to answer me but if ...
0
votes
0answers
32 views

closed-form solution for this constrained optimization

I want to find a closed-form solution for the vector $w=\left[\begin{array}{c} c\\-b \end{array}\right]$where $c$ and $b$ are column vectors, such that the following MSE is minimized: $\begin{align} ...
1
vote
3answers
51 views

Optimization fence problem with twist.

Suppose you have a 10x15 foot dog house and you wish to build a fence in a yard in a L shape to the north and east of the dog house. If you have 75 feet of fencing material available, what dimensions ...
0
votes
0answers
47 views

Weird optimisation problem

So, I have this problem: Let: $\mathbf{TB}$ be a matrix of size $t \times b$ ; $\mathbf{TC}$ a matrix of size $t \times c$ and $\mathbf{BC}$ a matrix of size $b \times c$; The matrix $\mathbf{BC}$ ...
1
vote
0answers
30 views

Maximum / Minimum Cost of a Box

this is a sample final question for a multivariable calculus course. "A rectangular box has two opposing sides (left and right) made of gold, two (front and back) of silver, and two (top and bottom) ...
1
vote
1answer
264 views

Linear Programming with One Quadratic Equality Constraint

I have a problem which can be formulated as a Linear Programming with One Quadratic Equality Constraint: where variable x is n-dimensional vector and H is a Semi-Positive Definite n-by-n matrix. I ...
1
vote
1answer
26 views

Multivariate function maximum criterion

Be a concave mutivariate function $f(\textbf{x})=\textbf{y}$. I observed the following conjecture: the maximum value of $f$ is achievable when all entries of $\textbf{x}$ are equal. How to prove such ...
0
votes
1answer
20 views

Probability density function / maximum likelihood for correlating sequence

I have a stream that contains two consecutive identical sequences, each of length $N$. These sequences have a ideal autocorrelation property. So I want to have the probability density function over ...
-1
votes
3answers
66 views

Maximize $xyz$ for $x,y,z>0$ satisfying $4xy+6yz+8zx=9$ [closed]

If $x,y,z$ are positive real numbers satisfying the equation $4xy+6yz+8zx=9$, then find the maximum possible value of $xyz$.
1
vote
2answers
15 views

Help with a homework problem involving $\textbf{H}$-conjugate vectors

My problem is the following: Let $\textbf{H}$ be a symmetric $n\times n$ matrix. Are the following claims true? Why? a) If the vectors $\textbf{d}_1$ and $\textbf{d}_2$ and vectors ...
4
votes
2answers
62 views

Proving the shortest function connecting two points is a straight line WITHOUT assuming the Euler-Lagrange equation

For learning purposes, I'm trying to prove that the shortest function passing through the two points $(x_1, y_1)$, $(x_2, y_2)$ is a straight line, without using the Euler-Lagrange equation. My ...
0
votes
2answers
50 views

Parameter optimization using a regression model.

I am working on an optimization problem. I build a regression model to understand the behavior of a system which depends on two variables which are functions of another two variables. My regression ...
1
vote
1answer
26 views

Maximum vertical distance between the line $y = x + 30$ and the parabola $y = x^2$ for $−5 ≤ x ≤ 6$

What is the maximum vertical distance between the line $y = x + 30$ and the parabola $y = x^2$ for $−5 ≤ x ≤ 6$? This is what I did but didn't work: Set $y_1=x+30$ and $y_2=x^2$, plugged ...
0
votes
1answer
14 views

Second Order Necessary Condition for Optimality

Question: [See context below.] What would be the analog of the Thm when $f$ is only defined on, say, a domain $D\subset\mathbb{R}^n$? In that case we can't take a general ...
0
votes
1answer
18 views

optimization with non smooth constraint

I am trying to maximize the profit of a power plant. I have a constraint which is that the power plant, when operating, has a minimum and maximum capacity. (So a power block either has an output of ...
1
vote
0answers
33 views

Can we ever have E(argmin(f)) = argmin(E(f))?

Consider a parametric real-valued function $f_{\boldsymbol{\alpha}}:\ \mathbb D^N \rightarrow\mathbb R$ whose parameters $\boldsymbol\alpha$ vary according to some distribution $\psi$, and $\mathbb D$ ...
0
votes
0answers
11 views

A textbook question from Fletcher's Practical Methods of Optimization

It is Q.2.19 on P.42: Given $q(x) = \dfrac{1}{2}x^{T}Gx+b^{T}x+c$ be a quadratic function, where $G$ is an $n \times n $ symmetric matrix and $b \in \mathbb{R}^{n}.$ (a) Show that a minimizer exists ...
1
vote
1answer
48 views

Optimization of a function

I need to optimize $$f(x,y,z)= x^2-y+e^{z}$$ with the restriction $$(x-2)^2+(y-3)^2+z^2=1$$ I've tried to substitute the restriction in $f(x,y,z)$ but it seems not to work. And when trying to use the ...
0
votes
0answers
10 views

Rank Minimization

I have a n*m matrix, the rank of matrix (r) is near to min(m,n) I want to minimize the rank by removing some of the rows or columns to get r << min(m,n) The goal is to achieve least rank ...
0
votes
0answers
15 views

Minimization of a weighted least-squares problem by Lagrange multiplier method

Problem: Let $Y = (y_1, y_2, \dots, y_m) \in \mathbb{R}^{m \times n}$ and $k \in \mathbb{R}^{m}$ satisfy $\sum_{i=1}^{m} k_i =1$ and $k \geq 0$. Show that $x=Yk$ is a minimizer for $h(x) = ...
0
votes
0answers
14 views

When can i solve simplex tableau

I saw exercises where they give an objective function ( without restrictions ) and a simplex tableau to be completed , if you can solve How do I know when it may solve the tableau ? What are the ...
0
votes
0answers
9 views

direct connection between gradient descent and follow the (perturbed) leader algorithm or weighted majority? [migrated]

Is there a direct conversion between gradient descent ([1], Alg 1 ) and any of the following algorithms? 1) Weighted Majority: http://onlineprediction.net/?n=Main.WeightedMajorityAlgorithm 2) ...
0
votes
0answers
32 views

Maximum area of a isosceles triangle in a circle with a radius r

As said in the title, I'm looking for the maximum area of a isosceles triangle in a circle with a radius $r$. I've split the isosceles triangle in two, and I solve for the area $A=\frac{bh}{2}$*. I ...
2
votes
1answer
61 views

Prove optimal solution to dual is not unique if optimal solution to the primal is degenerate

How do I prove an optimal solution to dual is not unique if an optimal solution to the primal is degenerate? I have no idea how to start this. Anyone know any books with these kinds of questions (and ...
1
vote
1answer
33 views

Optimization with Linear constraint $Ax=0$

I confront with this problem: $$\min_{x \in \mathbb{R}^{n}} \dfrac{1}{2} \left\| x- a \right\|_{2}^{2}$$ subject to $$Ax=0.$$ My tactic is to use Lagrange multiplier method that: $$\mathcal{L}(x, ...
1
vote
1answer
27 views

Selection of the mean of random variables to optimize the expected value of objective function

Here is the objective function to be maximized: $$ E_{v}(\log(1+v^{\mathsf T} \Lambda v) ) $$ where $v$ is a Gaussian distributed random variable vector $v ∼ \mathrm{CN}(M,I)$ with its mean vector ...
4
votes
2answers
140 views

Smallest value taken by a quadratic polynomial in two variables.

Let $p$ be a degree $2$ polynomial with integer coefficients, say $$p(x,y) = Ax^2 + By^2 + Cxy + Dx + Ey + F.$$ I would like to find an algorithm which solves the following: Problem 1: Given ...
2
votes
0answers
36 views

Transform a minimization problem to LP

This is a past examination question. I was asked (Q.1) to find an equivalent linear programming problem of: $$\min_{x \geq 0} \left \|Ax-a \right\|_{1} + \left\|Bx-b \right\|_{\infty}$$ where $A$ ...
0
votes
1answer
23 views

Will this optimisation give the global maximum?

My book optimises a type of functions using the lagrange method. From calculus I remember that we had to check the boundary when using lagrange, because it only gave local max, but it is not mentioned ...
1
vote
1answer
14 views

Find the Maximum and Minimum of the Given Function on the Given Plane Region

I've been good with most of the max/min finding in different regions, but this one's really messing with me. Can anyone lend a hand? Thanks. z = 2xy Region is the circular disk $x^2 + y^2 =< 1 $
0
votes
0answers
11 views

minimal sum of product of triples

I have a bunch of positive integers $a_1, a_2, ... a_{2n}$. I split these numbers into groups of 2, calculate the product of each pair, und sum over those. E.g. $Sum = a_1a_2 + a_3a_6 + a_4a_5$. I ...
0
votes
0answers
13 views

Polar cones' property [duplicate]

I am trying to prove: $A \subseteq B \implies B^\circ \subseteq A^\circ$ where $A^\circ$ is polar cone of $A$ ($A$ convex cone) and $B^\circ$ is polar cone of $B$ ($B$ convex cone)
2
votes
1answer
27 views

Slice an ellipsoid into equally thick slices for maximal surface

After seeing a colleague slicing a nearly ellipsoid piece of ginger for his cup of tea into almost equally thick slices to get more surface area (so the tea would suck out the ginger taste better), i ...
0
votes
1answer
264 views

De Jong's Fifth Function's Minimum?

What is the minimum solution to De Jong's fifth function, in the range $-65.536\leqslant x_1\leqslant 65.536, -65.536\leqslant x_2\leqslant 65.536$?