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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Finding Extremas of |x|

I'm trying to find the extrema of mod(x) but I'm not being able to do so. My attempt: f(x, y) = |x| f_xx = 0, f_yy = 0, f_xy = 0. So, D(x, y) = 0. And second derivative test isn't helping me at ...
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97 views

Partial derivative with matrices

I have reforumulated my problem of computing some quantities $\mathbf{a}\in R^{m}$ from $\mathbf{b}\in R^{n}$ in a matricial form: $$\mathbf{b} = (C\odot(\mathbf{1}_{n}\cdot \mathbf{a}^{T}))\cdot ...
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110 views

Equality case in elementary form of Holder's Inequality

A well known elementary formulation of Holder's Inequality can be stated as follows: Let $a_{ij}$ for $i = 1, 2, \dots, k; j = 1, 2, \dots, n$ be positive real numbers, and let $p_1, p_2, \dots, p_k$ ...
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61 views

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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49 views

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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52 views

linear programming slater condition

I am wondering if anyone could help to come up with a such example: ...
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43 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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The cylinder of maximum volume that can be drilled from a sphere of given radius [closed]

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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99 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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316 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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154 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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813 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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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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119 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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32 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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60 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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342 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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75 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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142 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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173 views

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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111 views

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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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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58 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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89 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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191 views

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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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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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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327 views

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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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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240 views

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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144 views

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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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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147 views

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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82 views

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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436 views

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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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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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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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 ...
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The optimization problem of soft margin Support Vector Machine: How to interpret?

I try to understand what exactly we are trying to optimize in the case of Support Vector Machine problem, which supports soft margins. The original problem is posed first as, without soft margins ...
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Show that weak local minimum of a convex function $\mathbb{R}^N\rightarrow \mathbb{R}$ is its weak global minimum.

Show that weak local minimum of a convex function $\mathbb{R}^N\rightarrow \mathbb{R}$ is its weak global minimum. Does the same happen to strong minimums? I know that when $f$ is convex, then we ...
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2answers
547 views

Does being a local minimum imply a positive definite hessian?

If $p\in R^{m}$ is a local minimum of $F:R^{m}\rightarrow R$, then can we conclude that $\dfrac{\partial ^2F}{\partial x \partial x'}[p]$ is positive definite? I guess you guys answers have ...
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Optimization - find the dimensions of a box as functions of volume - minimal surface area

Had a basic calculus course exam today. This was one of the problems: We have a rectangular box of a given volume V. Present the width, height, and length of the box as functions of V so that the box ...
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If we have $f(x)=e^x$, then what is the maximum value of $δ$ such that $|f(x)-1|< 0.1$ whenever $|x|<δ$?

If we have $f(x)=e^x$, then what is the maximum value of $δ$ such that $|f(x)-1|< 0.1$ whenever $|x|<δ$? I tried to solve this problem with delta-epsilon definition from the definition, 1 is ...
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54 views

conditions for $A +B$ to be semi-definite.

Suppose $A$ is a positive definite real matrix, and $B$ is symmetric and real matrix with $B_{ii}>0$. Are there conditions on $\sup_{j}|B_{ij}|$ that can guarantee $A+B$ is semi-definite. ...
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59 views

Using Lagrange multipliers to find the extrema of $f(x,y) = e^{2xy}$ subject to $x^2+y^2 = 16$

Find the maximum and minimum values of $f = e^{2xy}$ with respect to $x^2+y^2 = 16$. Using Lagrange multipliers, $\nabla f = \lambda\nabla g$. Therefore, the constraints are the following: ...