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)

1
vote
1answer
422 views

Absolute extrema

Find the absolute extrema of the function $f(x,y)=2xy-x-y$ over the region of the xy-plane bounded by the parabola $y=x^2$ and the line $y=4$ I was wondering if I needed to use Lagrange multipliers to ...
-1
votes
0answers
26 views

Maximum of function on field

My task is to find a maximum of function: $ f(x,y)=8x^2+2xy+2y^2-7x-4y-6 $ on filed: $ \{ (x,y); 9x^2+2xy+2y^2-5x-10y+12\ge0, x+6y\le14,2x+3y\le8,x\ge0,y\ge0 \} $ I started with just drawing out ...
0
votes
1answer
18 views

I am trying to maximize the following constrained optimization and I need help.

$$ \arg \max\limits_{C,D} \quad tr\{C^{-1}D\} + \log(det(C)) - \log(det(D)) \\ \mbox{sub. to} \quad tr\{C\} \le k \\ \quad \quad D > 0 $$ I did the following. Rewrite the above ...
0
votes
0answers
54 views

Maximum of a multivariable function

Let $0 \leq x,y,z \leq 1.$ Define the function $$f(x,y,z) = y^2 -2z^2 + 2x^2y. $$ What is the maximum of $f$ subject to the equality constraint $$ x = z + y\sqrt{(1-y^2)}?$$ Numerical methods give ...
0
votes
0answers
10 views

Critical points of quadratic form

Consider the problem $$ \text{max } Q(\mathbf{x})=\gamma_1x_1^2 + \gamma_2x_2^2 + \dotsb + \gamma_mx_m^2 \quad \text{ subject to } x_1^2 + x_2^2 + \dotsb + x_m^2 = 1 . $$ The $\gamma$'s are known and ...
1
vote
0answers
30 views

Local minima of a functions

Let the $f(x)$ is a strictly convex function on the $(a,b)$, and $\lim_{x\to a^+}=\infty$, $\lim_{x\to b^-}=\infty$, then exists only one local minima on $(a,b)$. Is this true? sorry for my English
0
votes
1answer
19 views

derivative free optimaization method

Currently I am working project on the derivative of free optimization methods. however, I want find practical problem that solved using this method. So, how can I get solve practical examples using ...
0
votes
3answers
24 views

An upper bound on sorting algorithms

I think I have a proof that $n\ln n$ is optimal for sorting algorithms. See here for a list. It must be greater than $n$ as this is too linear, and the $\ln$ factor comes from the harmonic series, ...
0
votes
0answers
19 views

Formulating an LP cost minimization problem

I am trying to solve this minimization problem using a software called GAMS.... The main difficulty I am running into is on how to formula this problem...I always have difficulty with word ...
0
votes
4answers
22 views

extrema with constraints (lagrange?)

I'd like to find the point of $E: 2x+3y+z = 14$ which has the smallest distance to the point of origin (0,0,0). I think I have $ d(x,y,z) = \sqrt{x^2+y^2+z^2}$ with constraint $2x+3y+z = 14$. What ...
7
votes
2answers
484 views

Graph theory problem (edge-disjoint matchings)

Find the smallest number $x$ so that if an $n$-vertex simple graph has at least $x$ edges then it contains $k$ pairwise edge-disjoint perfect matchings* ($k$ is a positive integer, $n$ is an even ...
0
votes
0answers
23 views

Optimization Toolbox (fmincon) - How to set logical constraints?

I'm pretty new to Optimization and barely understand it (was about ready to slit my wrist after figuring out how to write Objective Functions without any formal learning on the matter), and need a ...
2
votes
0answers
191 views

How to load warehouse pallets efficiently?

Assume that we would want to develop a warehouse management system, which picks up plastic boxes and stacks them on a pallet. A pallet has a maximum of $5$ vertical box stacks and the maximum height ...
0
votes
1answer
403 views

How to see that K-means objective is convex?

I'm trying to proof that the objective of the K-means clustering algorithm is non-convex. The objective is given as $J(U,Z) = \|X-UZ\|_F^2$, with $X \in\mathbb{R}^{m\times n}, U\in ...
3
votes
1answer
165 views

Prove $\exists\bar{b}$ s.t. every solution in polyhedron $P = \{x | Ax \ge \bar{b}\}$ is nondegenerate

I'm doing an exercise in the book "Introduction to Linear Optimization" and be stuck in this problem. Can anyone here help me solve this. I really appreciate all your help. Consider a ...
1
vote
1answer
129 views

Show that every extreme point in Q is either an extreme point of P or a convex combination of two adjacent extreme points of P

P is a bounded polyhedron in $\mathbb{R}^n$, $a$ a vector in $\mathbb{R}^n$, and $b$ some scalar. Define $$Q = {x \in P | a'x = b}$$. Show that every extreme point in Q is either an extreme point of P ...
0
votes
1answer
45 views

Gradient in mirror descent

Mirror descent can be in general written as \begin{equation*} \nabla\Phi(x_{t+1})=\nabla\Phi(x_t)-\lambda_t\nabla f(x_t), \end{equation*} where $f$ is the objective function and $\Phi$ is a convex ...
0
votes
0answers
13 views

Differentiating integral by substituting inverse function

I have the following cost function that I wish to minimize with respect to $\alpha$: ...
0
votes
1answer
439 views

Shortest distance between parallel line and plane

I've been doing questions regarding the shortest distance between lines/planes and points , and I've come across a question asking to find the shortest distance between a line and a plane which are ...
2
votes
0answers
150 views

Lagrange multiplier expression

I would like to solve the following optimization problem using the gradient ascend method: \begin{array}{ll} \text{maximize}_{\theta} & \theta^TQ_1\theta + b_1\\ \text{subject to} & ...
3
votes
2answers
76 views

Strategy for selecting the optimal time to check a cooldown timer

This is a hard problem for me to word in the title, so I'll try to do better now. Consider the following "game": You are sitting in a room beside a table. In the middle of the table there exists a ...
3
votes
1answer
102 views

An Extension to the Generalized Eigenvalue Problem

Given two square matrices $A,B \in \mathbb{R}^{n\times n}$, the generalized eigenvalue problem is finding the scalar $\lambda \in \mathbb{C}$ and vector $x \in \mathbb{C}^{n}$ such that $$ A ...
1
vote
0answers
63 views

Why covariance constraint subsumes the average power constraint?

I am studying an optimization problem in the form of \begin{equation} \begin{aligned} &\underset{p(x)}{\text{maximize}} & & W\\ & \text{subject to} & & 0 \preceq K_{X} ...
1
vote
0answers
45 views

Find max, min of 2D-function

I have obtained a result, but I'm not sure that it agrees with WolframAlpha results. Can you help me to understand? $$f(x,y)=e^{(x^2-y^2)/(x^2+y^2)}$$ The function is always increasing, so I can ...
0
votes
0answers
48 views

Optimization over a discrete set

For any given real numbers such that $\lambda_1\geq\lambda_2\geq\lambda_3\geq\lambda_4\geq\lambda_5\geq\lambda_6$, show that the optimal solution of the problem \begin{align} \mbox{maximize}& ...
0
votes
0answers
23 views

Secretary Problem modified for the stock market

In the standard version of the secretary problem, the chooser must select one secretary, the chooser is not allowed to go back and choose a rejected secretary and the number of candidates is known in ...
0
votes
3answers
33 views

What are the dimensions of the region that minimizes the quantity of fence?

A Farmer wants to a fence a rectangular region of $600m^2$ and then divide the middle of it also with a piece of fence that is parallel to one of the sides of the rectangle. What are the dimensions ...
0
votes
0answers
23 views

Constrained Optimization when unconstrained gradient is known

I have a function $f(a_1, a_2, a_3)$. Inside the function the parameters are transformed into \begin{equation*} w_1 = (a_1+a_2+a_3)/3 \\ w_2 = (a_1+a_2+a_3)/3 - a_1 \\ w_3 = (a_1+a_2+a_3)/3 - a_2 \\ ...
7
votes
1answer
135 views

What function maximizes area for a constant arc length?

Suppose I have a continuous function $f$, such that $f(0) = f(1) = 0$. Given the length $l$ of the curve between $0$ and $1$, which function maximizes the area under the curve? I know that if $l \leq ...
0
votes
0answers
63 views

How to find minimum distance path between 2 points on a surface

Given a surface equation is $z=f(x,y)$ and also given two points on surface are $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$ How can be found the path equation $(x=p(t),y=s(t),z=u(t))$ that it creates ...
3
votes
2answers
43 views

Taylor theorem equation

I have one question about Taylor theorem. Originally, Taylor theorem is represented as $$f(x) = f(a) + f'(a)(x-a) + \ldots$$ But my book says Suppose that $$f : R^n → R$$ is continuously ...
0
votes
1answer
39 views

static offline bipartite graph matching

Consider a static offline bipartite graph where we have complete knowledge of the two sets of vertices $U$ and $V$. Now an edge is drawn between a vertex of $U$ and vertex of $V$ if the difference of ...
2
votes
1answer
44 views

Minimize $w=9y_1+4y_2$ subject to linear inequalities

Minimize $w=9y_1+4y_2$ subject to : $4y_1+9y_2\geq 360$ $y_1+4y_2\geq 40$ $y_1\geq 0,~y_2\geq 0$
1
vote
1answer
34 views

Scaling in utility maximisation

If I have the wealth process $$dw_t=rw_tdt+n_tS_t(\sigma dB_t+(\mu-r)dt)-c_tdt,$$ where $n$ is number of $S_t$ and $B_t$ is Brownian motion. If we define the admissible set $A$ as follows: ...
1
vote
0answers
38 views

Derivation of Von Karman Equations

I'm reading Howell's Applied Solid Mechanics to gain background for a research project. I'm struggling with the following derivation in the text that the authors refer to as a "lengthy exercise." The ...
1
vote
0answers
12 views

What is the Euler Lagrange condition for SDEs?

Does the Euler Lagrange condition... $$\frac{d}{dt}\left( \frac{\partial L}{\partial \dot{x}}\right)-\frac{\partial L}{\partial x}=0$$ ...have a meaningful extension to Stochastic Differential ...
0
votes
0answers
11 views

Maximizing the weighted sum of two CDFs subject to a constraint on the expected value.

I encountered this problem in a proof and would like to have your help: Consider the maximization problem: \begin{eqnarray} \max_{x,y}b_x\Phi(x)+b_y\Phi(y),s.t\\ ...
3
votes
0answers
30 views

What does it mean for a problem to be time-homogenous?

(This is an associated question to Scaling in utility maximisation. $c_t$, $w_t$, $n_t$, $A$ are defined there.) I am reading that because of time homogeneity $$\sup_{(n,c)\in ...
1
vote
2answers
61 views

How to find max value without Lagrange

I am trying to find the maximum and minimum values of the function $$f(x,y,z)=2x-y+4z$$ on the unit sphere $$x^2+y^2+z^2=1$$, but without using langrange multipliers or gradient. I would like to do ...
2
votes
2answers
56 views

Image restoration in matlab via PDE toolbox

I want to remove a noise for an image using matlab, when the observed image is $$f=u+v$$ where $u$ is the restored image (is the image i want recovered) and $v$ is the gaussian noise. To restore $u$, ...
0
votes
0answers
15 views

When is the likelihood function concave?

Is there a particular criteria that let's us determine whether the likelihood function of a function is concave? I'm dealing with the EM Algorithm and since there we only find local maxima, I was ...
1
vote
1answer
14 views

Dual variable calculus

If $z=v'(w)$ and we introduce new variable $J(z)=v(w)-wz$. Then it is clear that $J'(z)=-w$ but why is $J''(z)=-1/v''(w)$?
3
votes
0answers
30 views

Shortest system of roads between 4 cities

You have $4$ cities placed on the vertices of a square of side length $1$ km. You have to come up with a system of roads such that you can reach any city from another (directly or through another ...
2
votes
0answers
31 views

What subjects properly belong in operations research as their “owning” discipline?

Warning: This is a soft question, hence I would make it a wiki-community post if I could. Operations Research involves a broad swath of disciplines, ranging from probability and statistics/stochastic ...
5
votes
1answer
70 views

Polynomial between $0$ and $1$ that produces largest integral

Question: Let $n\in \mathbb{N}$. Find the polynomial $p(x) = \sum_{i=1}^n a_ix^i$ that satisfies $p(1) = 1$ (and $p(0)=0$ since we already have $a_0=0$) $p(x) \in [0,1]$ for all $x\in ...
0
votes
0answers
33 views

Optimum point of $f(s) = \int_0^{\pi} \frac{ \exp(-s) y \cos(ky)}{s^2+y^2} \,dy $

Is it possible to find optimum point for the following function f(s) (i.e. $df/ds=0$): $$ f(s) = s e^{-s} \int_0^{a} \frac{ y \cos(\frac{\pi}{a} y)}{s^2+y^2} \,dy $$ or $$ f(s) = s e^{-s} ...
2
votes
1answer
90 views

What would be the objective functions for this problem?

I have the following data (this is just a sample of my entire dataset): # Distance PriceIndex Rating 1 400 3 5 2 420 2 4 3 500 1 2 Considering the ...
1
vote
1answer
44 views

Finding the smallest square inside a parabola. [duplicate]

I just thought of a problem earlier today, but wanted to know if there was an easier way of acquiring the answer. Say I have a standard parabola $y=x^2$ with 3 points on it $P,Q,R$ and another point ...
0
votes
0answers
11 views

Dropping Multiple Constraint in Active Set Optimization

Does anyone know an example for the Exercise 2.5.2 of Bertsekas nonlinear programming book?: Show by example that in the method of this section, if several constraints $a_j'X \le b_j$ with ...
2
votes
1answer
73 views

Algorithm - Maximum subarrays with sum and OR

I was thinking on the following problem: Given an array A. The value of an interval from i to the index j is defined as follows: Take the maximum value from that interval, and add it to the OR ...