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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How do you compare carsharing plans to calculate the cheapest?

Call hourly rate = HR. Assume that I can guess my monthly usage in hours, which I call $g$. Beware that the fixed fees are presented in different units of time, so first convert everything into ...
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Optimization 101 for electrical engineers…Where to start from?

I have never taken any optimization class. From an electrical engineering point of view, how should I approach learning this field? What kind of information I should be looking at in my problem to ...
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linear programming slater condition

I am wondering if anyone could help to come up with a such example: ...
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39 views

Get the minimum value for multivariable

I need a way to calculate the very minimum of each variable for a grade average. Each grade variable have a weighing percentage (I don't know if it's the right term) The weighing sum must be the ...
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15 views

Uniqueness of the minimiser of a convex functional

It is well known that a lower semicontinuous and convex functional possesses its own unique minimizer. How about such minimization problem for the functionals as follows? $ \min_{x \in X}\{F(x)\} = ...
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19 views

local and absolute max and min of the least integer function

how can we find the local and absolute maximum and minimum value of the least integer function.I thought that theta won't be a local max and min as this function has the same value over a particular ...
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The cylinder of maximum volume that can be drilled from a sphere of given radius

Find the dimensions of the cylinder whose maximum volume can be drilled from a steel sphere of radius 8. As Amanda Kelius said, my problem is how to setup the expression for the volume. I know ...
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57 views

Trouble with formulation of objective function (constraint optimization)

I am new to optimization and I will try to state my question as clear as I can. I need to solve this constraint optimization problem. I want to find real vectors $\mathbf{f}$ and $\mathbf{g}$ that ...
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72 views

Nested optimization problems solving using mixed integer linear programming

Let us have two vectors of decision variables, $\mathbf{x}$ and $\mathbf{y}$, two linear objective functions, $F \left( \mathbf{x}, \mathbf{y} \right)$ and $f \left( \mathbf{x} \right)$, and two sets ...
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25 views

Set similarity metrics

I have two sets $A$ and $B$. Each of them has a set of instances. Say $A = {a,b,c,d}$ and $B = {b,f,g}$. What are the most performing metrics out there to compute the similarity between the two ...
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71 views

Conditional inequalities

Let a,b,c be positive real numbers such that $abc=1$. Prove that $$\frac 1{a^3(b+c)}+\frac 1{b^3(c+a)}+\frac 1{c^3(a+b)} \ge \frac 32$$ We can derive the following inequalities from the given equality ...
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25 views

Gauss-Newton vs Gradient descent

I would like to ask first if the second order gradient descent method is the same as the Gauss-Newton method. There is something I didn't understand. I read that with the Newton's method the step we ...
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36 views

Add vectors from a set to reach the goal vector, using the minimum possible cost

I am trying to solve a problem in an optimal way. The problem is as follows: We have an n-dimensional space In this space, we have a "finish" point with n coordinates, all non-negative We have a set ...
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108 views

Given 500 parts and a list of orders, pick 50 parts to maximize the number of fulfillable orders

I'm going to start with a proclamation that this kind of optimization is new to me, so don't fault me for setting up the problem in a weird way. Please let me know if this is unclear. In a ...
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22 views

Proving that two problems are strongly dual when solutions are restricted to a space

Consider the following problems with solutions $\mathbf{w}\in\mathbb{R}_{++}^n$ \begin{align} (P) \hspace{.3in} \min_{\mathbf{w}} \hspace{.3cm} & \mathbf{p}^H\cdot\mathbf{w} \\ \text{s.t. } & ...
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21 views

Job scheduling to minimise squared completion times using mixed 0-1 quadratic program

I have come across an Optimization question as follows: There are $n$ jobs that have to be processed on a machine. The machine can process only one job at a time. The time taken to process job ...
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43 views

Maximum time-to-exit of random walk in R^n

I am trying to solve the following problem : Given a set $A$ in $\mathbb{R}^n$ and a point $p$ , I want to find a convex subset of $A$, call it $C$, such that $p$ is in $C$ and random walk starting at ...
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24 views

IF a cone is inscribed in a larger cone,then what will be the radius of the small cone if it has the maximum volume?

If a smaller cone is inscribed in a larger cone as shown, then what will be the radius of the smaller cone if it has the maximum volume? Attempt I know that the volume of a cone ...
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36 views

Non-decreasing Convex function?

When my textbook states, "Non Decreasing Convex Function", does it mean that the function is convex and increases in y for every x from its minimum? That is if f(x) = y is convex. Please explain if ...
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15 views

Literature for Load Balancing Problems

i'm am looking for literature. i found some but there might be a better name to search for. It's like load balancing for machines to minimze the biggest concurrent load. Now you have a time aspect. ...
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81 views

Strong duality for nonconvex quadratic program (with multiple constraints)

Consider the following optimization \begin{eqnarray} P_1: \quad &\underset{x\in\mathbb{C}^N}{\mathrm{minimize}}&\; f_0(x) \\ &\mathrm{subject\;to}&\; f_i(x) \leq 0, i=1,\ldots,m \\ ...
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solving LP problem : no optimal solution exists?

$$\max[Z(x,y)=3x+2y]$$ $$-x+y\le 1$$ $$-x+2y\le4$$ When I tried to solve the above maximization LP problem using the simplex method, from the first iteration, all basic variables became negative. ...
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SOS relaxations for polynomial optimization

I do not understand how SOS (Sum-Of-Squares) relaxation for polynomial optimization works in some cases. For instance, consider the polynomial optimization problem: \begin{equation} ...
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Structural / design / meta optimization - is there mathematical theory. Optimization over categories?

There is huge branch of mathematical optimization theory, but it mostly considers the finding optimal parameter values for the predefined structures. There are variational calculus and optimal control ...
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How to understand ' Let $\mathcal{H}$ be a Hilbert space of functions $f$ : $ \mathcal{X} \rightarrow R$, denoted on a non-empty set $\mathcal{X}$.'

I am a beginner. By asking this question, I means that, to construct a Hilbert space, should $\mathcal{X}$ satisfy some properties? Furthermore, in some papers especially on machine learning, ...
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Global optimality of a convex but non-smooth function

I have a question. The answer may be too obvious but I cannot be sure about the right answer. Let say that we have a convex but non-smooth function which is defined as $f : \mathbb R^2 → \mathbb R$. ...
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How to reconstruct a sparsely sampled multiperiodic function?

I have $m$ oscillators, where $m$ is unknown, with periods $\vec p = p_1, p_2, \ldots, p_{m}$. Each of the oscillators $j$ has associated with it a vector of sine coefficients $\vec A_j$ and angle ...
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23 views

How to solve an optimization problem where the size of the solution is part of the objective

I want to find the smallest vector $\vec p$ such that some constraints are satisfied, so something like: $$\hat p = \underset{\vec p}{\arg \min} \; |\vec p| \\ s.t. \; F(x_i, \vec p) \leq \epsilon_i ...
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53 views

Maximum of $f(x) = (45-2x)\cdot (24-2x)\cdot (2x)\;,$ Where $0<x < 12$

How Can I Maximise $f(x) = (45-2x)\cdot (24-2x)\cdot (2x)\;,$ Where $0<x < 12$ Using Inequality $\bf{My\; Try::}$ In $0<x<12\;,$ The value of $(45-2x)\;,(24-2x)\;,2x>0$ and we can ...
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47 views

Maximize ratio of logarithms

How can one maximize the ratio of two logarithms $ \frac{\log{f(x)}}{\log{g(x)}}$ where the argument to each logarithm is the (positive) ratio of two first-degree polynomials? I have tried ...
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Dynamic programming problem for discrete linear time varying system

I'm working on a linear time varying discrete(LTV) multi input multi output(mimo) system. I formulate the problem description in the following way $$x_i(k+1) = x_i(k)\cdot A_i(k) + B_i(k)\cdot ...
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Geometric Intuition behind the Dual Norm?

What is the geometric intuition behind the dual norm, $\|z\|_* = \sup \{ z^Tx \text{ } | \text{ } \|x\| \le 1\}$ Specifically, if possible in terms of hyperplanes defined by $x$ and $z$. My interest ...
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The Convergence of Coordinate Descent involving multiple variables

Given a convex, but not differentiable function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ can be decomposed into two parts, namely, $f(x) = g(x) + \sum_{i=1}^n h_i(x_i)$, where $g$ is convex and ...
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Farewell 2014 welcome 2015 - “Math Golf” [closed]

In programming, Code Golf is a competition in which the participants are trying to implement an algorithm with code which is as short as possible. Also, Stack Exchange has a site dedicated to Code ...
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How to deal with a very small line search step in optimization?

The Armijo type line search is to find an $a_k > 0$ such that $$ f(x^k + \alpha_kd^k) \leq f(x^k) + \sigma_1 \alpha_k \nabla f(x^k)^Td^k $$ given $\sigma_1 \in (0, 1/2)$. We know that for ...
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Linear programming algorithm that minimizes number of non-zero variables?

I have real world problems I'm trying to programmatically solve in the form of $$Z = c_1 x_1 + c_2 x_2 + \cdots + c_n x_n$$ Subject to \begin{align} & a_{11} x_1 + a_{21} x_2 + \cdots + a_{n1} ...
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29 views

comparing optimizer of two functions

I know that there is a result that says: if $f^{\prime} (x) \geq g^{\prime}(x)$, then the maximizer of $f$ is $\geq$ maximizer of $g$ (and indeed I can prove it), I am wondering whether the result ...
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From constrained to unconstrained maximization problem

I have the following constrained maximization problem $$ \max_{X_1,X_2,...,X_i,...,X_N} \sum_{i=1}^{N}X_i f_i(X_1,...,X_N) \hspace{0.2 cm} \text{subject to} \sum_{i=1}^{N}X_i-B\leq 0 \text{ and } ...
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KKT conditions for non-convex almost everywhere differentiable problems

Consider the context of constrained function minimization. The well-known KKT conditions do not require the objective or constraint functions to be convex, but they do require them to be ...
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Wolfram alpha error in global minimization?

Let us consider the function $$f(x,y)= x + y^2 - \ln(x+y)$$ If you try to minimize it using Wolfram Alpha (http://www.wolframalpha.com/input/?i=minimize+x%2By%5E2-ln%28x%2By%29), it founds a local ...
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What is the complexity of Simplex Method's Phase 1?

What are the average and worst-case complexities of the Phase 1 of the Simplex Algorithm? Is it respectively polynomial and exponential as well? Google search did not yield any results unfortunately. ...
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short-sale constraint with nonpositive-definite matrix in portfolio optimization

This question is about portfolio optimization in R. I have a nonpositive-definite matrix. I have handled with the singularity. Unfortunately, quadprog etc. optimization packages fail to solve the ...
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How does one generate recognizable point-patterns on a plane?

I've recently learned that some smartpens (e.g. Livescribe) have a camera in their front part. They film the paper. You have to use special paper which looks as if somebody made a lot of tiny holes ...
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Normal Cone of $\mathbb{R}^n_+$ and $S^n$?

I'm trying to solve the problem $\min_x \{f(x) + \delta_X(x)\}$ where $f$ is a differentiable function and $\delta$ is the indicator function $\delta_X(x) = \begin{array}{l}0, x \in X \\ ...
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How do I solve the following equations?

I have the following problem: \begin{align} Y &= A X \\ Y &= R \exp \left(j \Phi\right) \text{element-wise}\\ X, R, \Phi &\in \Bbb R \\ A, Y &\in \Bbb C \end{align} I know what A is, ...
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Solving min-max optimization problems in original ways (that is, avoiding the frenzy of differentiation)

As I see from the students I'm tutoring, once faced with a min-max problem, the average student is taken by the frenzy of differentiation. I would like to show that sometimes it is better to use ...
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1answer
46 views

What is a simple proof that something is np complete that does not use np completeness of something else?

What is a simple proof that something is NP complete that does not use NP completeness of something else? Every proof seems to reduce to something else being NP complete.
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30 views

Find optimal matrix to maximize the expected expression

I am interested of the following function with $Q$. $$f(Q)=h^TQh-\frac{1}{(h^TQh+1)^2}-g^TQg+\frac{1}{(1+g^TQg)^2}$$ where $h$ and $g$ are both given $N\times 1$ vectors. And $Q$ is $N \times N$ ...
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Find max cone inscribed parallelepiped volume

There is a right cone with H = 20 and R = 12. How to find the best inscribed parallelepiped parameters which would provide its ...
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58 views

Maximising the Area of a Cyclic Quadrilateral

In cyclic quadrilateral $ABCD$, $AB = AD$. If $AC = 6$ and $AB/BD = 3/5$, find the maximum possible value of $[ABCD]$. (Source: SMT 2014) If we let $AB=AD = 3x$ and $BD=5x$, from Ptolemy, we have ...