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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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 ...
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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 ...
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4answers
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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 $x_0$, $f(x_0)'' < 0$ ?
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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 ...
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216 views

Maximal minimum for a sum of two (or more) cosines

Please prove (or disprove, and give the correct answer): $$2 =\mathrm{argmax}_{r\geq 1}\min_{x\in \mathbb{R}}\left[\cos\left(x\right)+\cos\left(rx\right)\right] $$ In other words, find $r \geq 1$, ...
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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 $y$, ...
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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 ...
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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 ...
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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 ...
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31 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 ...
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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. ...
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1answer
35 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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21 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 ...
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1answer
37 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. ...
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1answer
84 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 $\sqrt{y}.$
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19 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 ...
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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 ...
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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 ...
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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
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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 ...
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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 ...
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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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Differentiating integral by substituting inverse function

I have the following cost function that I wish to minimize with respect to $\alpha$: ...
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46 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 ...
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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 ...
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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 ...
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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}& ...
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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 \\ ...
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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 ...
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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 ...
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How to solve $\min \limits_{\mathbf{x}} \| \mathbf{Ax}-\mathbf{b} \|^2$?

Let $\mathbf{x}=[x_1,\ldots,x_K]$. I have the following optimization problem: \begin{array}{rl} \min \limits_{\mathbf{x}} & \| \mathbf{Ax}-\mathbf{b} \|^2 \\ \mbox{s.t.} & x_k\ge 0, \forall ...
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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 ...
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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 ...
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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\\ ...
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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 ...
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66 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 ...
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16 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 ...
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1answer
44 views

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

Minimize $w=9y_1+4y_2$ subject to : $4y_1+9y_2\geq 360$ $y_1+4y_2\geq 40$ $y_1\geq 0,~y_2\geq 0$
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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 $J''(z)=-1/v''(w)$?