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
12 views

Property of the local solution of a static constrained optimization problem

In Nocedal/Wright's Numerical Optimization (1999, 1E) on p. 332 in subsection Feasible Sequences of section 12.3 Derivation of the First-Order Conditions they claim that a local solution $x^*$ to a ...
1
vote
0answers
26 views

Asset optimization using Excel

I originally posted this in superuser, but was told it was more of a math problem than an Excel problem. I'm working on a project, part of which involves figuring out the number and type of vessels ...
0
votes
0answers
2 views

What is the behavior of the spatial median in high-dimensional spaces?

I am a photographer who is investigating a technique known as image stacking, in which multiple images of the same subject are combined to reduce noise (by CLT). Commonly used techniques are mean and ...
1
vote
0answers
48 views

Can I clamp singular values of $3\times3$ matrix without effectively computing SVD?

I have a $3\times3$ matrix $A$, and compute its SVD $U \Sigma V^\star = A$. I clamp the singular values in $\Sigma$ to some small range (e.g. $[0.5, 1.5]$ ) and reconstruct matrix $\widetilde{A}=U ...
1
vote
2answers
319 views

Gradient descent vs. Newton's method — which one requires more computation?

Broadly speaking, when numerically minimizing a d-dimensional objective function: Gradient descent generally requires more iterations, but each iteration is fast (we only need to compute 1st ...
3
votes
4answers
1k views

How to compute Lipschitz Constant for multivariate function $f(x,y)=1-xy$?

How to compute Lipschitz Constant for multivariate function $f(x,y)=1-xy$? I know the definition for one variable? What is its definition for multivariate functions?
0
votes
3answers
41 views

optimize the volume of a box where the sum of the h*d*w is the only known variable

I'm currently stuck with the following optimization excercice: Consider a box such that sum of its sides is $210$. Find the maximum volume that the box can have. Here are my current thoughts: ...
0
votes
1answer
20 views

Minimizing nonsmooth single variable functions?

What options is available if one wants to minimize a nonsmooth convex function of one variable? Subgradients would work, but there has to be some nice way of utilizing that we're only searching in 1d. ...
-1
votes
1answer
37 views

Finding min and max under constraints

I have a two variable function: $f(x,y)=4x^2-y^2-xy-2x+6y$. I need to find its absolute minimum and maximum under the constraints: $y=4-2x$, $x \geq 0$, and $y \geq-2$. I am not sure how to do it, ...
0
votes
0answers
18 views

Optimization methods to find valleys in a map

I have a map of some size say $1000\times1000$ pixels that is in a equivalent sized array. Instead of searching the map for a global minimum what I'd like to do is find a cluster of connected minimums ...
1
vote
2answers
34 views

General process to find global extrema of a function?

I have been reading and watching videos about local and global extrema, but all of this material covers the topic just graphically, and nobody really explicitly cares on how to find the global maximum ...
0
votes
0answers
17 views

Optimization approaches and algorithms for highly dimensional complex functions.

I am searching for an approach or algorithm that is best suited for finding a minimum or maximum in a n-dimensional function. All variables are complex, the function itself returns a real number. The ...
1
vote
1answer
131 views

How to minimize $a \times b$ where $a^b≥x$?

For example, if $x$ is 1 billion, the smallest possible $a \times b$ will be $3 \times 19 = 57$. This is because: $2^{30} \ge 1000000000$ $2 \times 30 = 60 $ $3^{19} \ge 1000000000$ $3 \times 19 ...
0
votes
2answers
395 views

Optimization, rectangle inscribed inside arch of the curve.

A rectangle is to be inscribed under the arch of the curve $y = 4\cos(0.5x)$ from $x = \pi$ to $x = -\pi$. What are the dimensions of the rectangle with largest area, and what is the largest area? ...
0
votes
0answers
24 views

Making projected search directions conjugate

I'm trying to implement a minimization process for the optimization problem: ...
1
vote
1answer
29 views

Finding Extreme Values (Multivariable)

Given $f(x,y)=x^2+2y^2$, find its extreme values on $x^2+y^2=1$. I know how to solve this problem using Lagrange's method and the constant variation method. The solutions are $(\pm1,0)$ and ...
3
votes
2answers
34 views

Regarding Max flow problem ( Ford-Fulkerson Algorithm)

I'm looking for the max flow in this graph but something is going wrong. First I take the path : 1-2-4-6. So the flow ie $F=1$ Then : 1-3-2-5-4-6 and the flow updates to $F=2+1=3$ If i take the ...
1
vote
1answer
17 views

quadratic programming problem with positive constrains

Is there a non-iterative solution to the following quadratic programming problem with constrains? Is there any problem to think the variable as some square of another variable to get ride of the ...
2
votes
2answers
41 views

Finding an optimal sequence

It's my first time on this site:) I have to find a strictly increasing finite sequence $\{x_k\} _{k=1, \dots, n}$ with $x_1=c^2$ that will minimize the following expression ...
1
vote
0answers
35 views

Modelling a warehouse in optimization?

I am trying to model the following rather general optimization problem. Let $p_{t}$ be a given non time series of product prices. These are fixed points $p_{t}$ is not described as a random variable. ...
1
vote
2answers
46 views

Find minimum value of $|2z-1|+|3z-2|;\,\,z\in\mathbb{C}$?

I tried this question using many different ways (triangle inequality, geometric interpretation, etc) but I didn't get the correct answer. The minimum value of $|2z-1|+|3z-2|;\,\,z\in\mathbb{C}$ ...
0
votes
0answers
18 views

Classify extreme points of multivariate implicit functions when cross derivative is not available

I have the following problem: Let $f(x,y)$ be a function defined on $[0,1]^2$ I want to prove that $f(x,y)$ has no local minimum for $x>y$. I have no idea about the sign of the cross derivatives ...
0
votes
0answers
28 views

Why we should study “sub-energy” function and what are its applications in mathematics?

I read some papers in optimization and I found this term "Sub-energy function" that is a new mathematical term to me. Aside from "its a large area in mathematics," why we should study it? What's its ...
1
vote
2answers
44 views

Projection of a vector onto the null space of a matrix

I have the following optimization problem: $$ \text{minimize}_x \Vert z - x \Vert^2 \\ \text{subject to } Ax = 0, $$ where $x,z\in \mathbb{C}^N$, and $A\in\mathbb{C}^{M \times N}$. $A$ is a wide ...
188
votes
16answers
7k views

Optimizing response times of an ambulance corp: short-term versus average

Background: I work for an Ambulance service. We are one of the largest ambulance services in the world. We have a dispatch system that will always send the closest ambulance to any emergency call. ...
0
votes
0answers
31 views

Prove a convex function: $F(a,u,c)=\int_{\Omega}u(x)\left (\log(a)+\left(\frac {f(x)-c}a\right)^2 +const\right )dx$

I have a function and I would like to know the function whether convex or non-convex. Let look at my function $$F(a,u,c)=\int_{\Omega}u(x)\left (\log(a)+\left(\frac {f(x)-c}a\right)^2 +const\right ...
0
votes
1answer
11 views

Local extrema of a function, considering its $n-th$ derivative where $n$ might be odd or even.

I found the following in my notes and need help to understand it. Consider the following theorem: "Let $A\subseteq \mathbb R^n$ be open, $f:A\to \mathbb R$. Suppose also that $f$ has all first order ...
2
votes
4answers
70 views

Is the opposite of the Second Derivative Test also true?

Given the Second Derivative Test, one case says : If $f(x_0)''<0$, then $f$ has a local maximum at $x_0$. Is it also true that, if $f$ has a local maximum at $x_0$, $f(x_0)'' < 0$ ?
0
votes
0answers
56 views

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 ...
7
votes
2answers
317 views

System $a+b+c=4$, $a^2+b^2+c^2=8$. find all possible values for $c$.

$$a+b+c=4$$$$a^2+b^2+c^2=8$$ I'm not sure if my solution is good, since I don't have answers for this problem. Any directions, comments and/or corrections would be appreciated. It's obvious that ...
1
vote
1answer
35 views

Operations Resarch Optimal Scheduling

Consider the following problem: A car manufacturing company needs to transport car frames, which are $10$ cubic units each, and wheels, which are $2$ cubic units each, across the Atlantic ocean. ...
2
votes
1answer
36 views

Split number into minimum sum components

I was wondering if there is an analytical solution for the following optimization problem? We have a given real number say $k$. It is needed to split $k$ into minimum number of real components, so ...
3
votes
2answers
975 views

Find maximum and minimum of $f(x,y) = x^2 y^2 - 2x - 2y$ in $0 \leq x \leq y \leq 5$.

Find maximum and minimum of $f(x,y) = x^2 y^2 - 2x - 2y$ in $0 \leq x \leq y \leq 5$. So first we need to check inside the domain, I got only one point $A(1,1)$ where $f(1,1) = -3$. and after ...
-1
votes
1answer
30 views

Maximise probability of non-empty urns by addings balls.

I have K urns and in each of them i have already some white and some black balls (different number in each of the urns). I have an equal chance of picking any of the urns. I have in my hands X white ...
3
votes
1answer
86 views

What more can be said about $\max_{v^\mathsf{T} v=1} \frac{v^\mathsf{T} B v}{v^\mathsf{T} A v}$?

Assume we have a positive semidefinite matrix $A$. Another matrix $B$ is equal to $A$ except it's $i$th row and$i$th column is zeros and element $B_{ii}=(n-1)A_{ii}$. i.e. \begin{align} B&=A-e_i ...
0
votes
1answer
27 views

I dont understand this statement: Prove that continuity in $x$ of the Gateaux derivative implies Frechet differentiability

Prove that continuity in $x$ of the Gateaux derivative implies Frechet differentiability I don't understand this statement, since Gateaux derivative is a function $f(x;y)=a\cdot y$ for all $y$, ...
0
votes
5answers
92 views

How to show that $\nabla \|x\|=\frac{x}{\|x\|}$, $ 0\neq x\in\mathbb{R}^n$

How to show that $\nabla \|x\|=\dfrac{x}{\|x\|}$, $ 0\neq x\in\mathbb{R}^n$. I can't use the partial differentiation since I don't know if it is differentiable, I have to use the definition, i. e. ...
0
votes
0answers
17 views

Minimizing the “distance” between a finite set of elements in a finite length sequence.

Given a set of "options", {A,B,C,C}, I'd like to construct a certain kind of sequence of these elements. And example sequence would be: ABCDABCD I define some average "distance" for this sequence ...
1
vote
0answers
22 views

Let $f$ be Gateaux differentiable and and $f'(x;y)$ is continuous at $x$. Show that $f$ is Frechet differentiable at $x$ [duplicate]

Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$ be a function such that is Gateaux differentiable and $f'(x;y)$ (the Gateaux derivative) is continuous at $x$. Show that $f$ is Frechet differentiable at ...
0
votes
1answer
519 views

Finding the center of mass of a cylinder

Help finding center of mass of soda can? If you represent the soda can as a right-circular cylinder radius=$4$ cm height =$12$ cm We are told to neglect the mass of the can itself. When the can ...
0
votes
0answers
17 views

Intuition on primal convergence in dual subgradient method

It is well known that the subgradient method applied to the Lagrange dual of a convex problem may produce a sequence converging to the dual optimum, but the primal iterates produced by this sequence ...
2
votes
1answer
35 views

maxima and minima piecewise function

I'm exercising on maxima and minima, I think I got the point of global and local extremes but then I find this piecewise function where my teacher says that the right answer is "c". I thought the ...
0
votes
2answers
20 views

Dual subgradient method - can we solve approximation of dual?

Consider the problem to minimize $f(x)$ under the constraints $x \leq b$ and $x \in X$. I Lagrange relax the constraint $x \leq b$ getting $L(x,u) = f(x) + u^t(x-b)$. When using the subgradient ...
1
vote
1answer
34 views

Let $f$ such that $\lim_{\varepsilon\rightarrow 0^+}\frac{f(x+\varepsilon y)-f(x)}{\varepsilon}=b+a\cdot y$ $\forall y$. Show that $b=0$.

Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$ be a function such that $$\lim\limits_{\varepsilon\rightarrow 0^+}\dfrac{f(x+\varepsilon y)-f(x)}{\varepsilon}=b+a\cdot y$$ $\forall y\in\mathbb{R}^n$. Show ...
0
votes
0answers
18 views

Conic Optimization

In a recent paper, Conice Optimization via Operator Splitting and Homogeneous Self-Dual Embedding, a primal of the form \begin{alignat}{3} &\text{minimize} &&c^T x\cr ...
0
votes
1answer
24 views

Definition of differentiation of scalar functions

I was reading the book of optimizaion by Polyak and I found this definition: A scalar function $f(x)$ of an $n$-dimentional argument $x$ ($f:\mathbb{R}^n\rightarrow\mathbb{R}$) is said to be ...
1
vote
0answers
20 views

How to find the maximum point of a function with four dimensional variable.

There is a very complicated non-concave but smooth enough function $f(X)$, where $X=(x_1,x_2,x_3,x_4)$. I want to find the maximum point of $f$ on a constrained set as follows: \begin{eqnarray}max ...
1
vote
0answers
35 views

Matlab Optimization problem with Matrices

I'm trying to solve an optimization problem in Matlab. The expressions you will find below. Problem is it is all matrices, and I have no idea which solver to use for that. $w$ is of size $n \times 1$, ...
3
votes
1answer
83 views

Maximum of $x+y$ with constraint

What is the maximum value of $x+y$ given that $x^2-4xy+4y^2+\sqrt{3x}+\sqrt{3y}-6=0$? $x,y$ are real numbers. Notice that it has terms $\sqrt{x}$ and $\sqrt{y}.$
1
vote
0answers
43 views

Quadratic optimization problem (inner products) with stochastic constraints

Let the set of feasible solution be the set of all row-stochastic $n \times k$ matrices $P = [p_{ij}]$, that is $\mathcal{P} := \{P \in \mathbb{R}^{n \times k} \ | \ P \mathbf{1} = \mathbf{1}, P \geq ...