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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Converges extremely slowly, using Douglas-Rachford splitting, how to improve?

my problem looks like this: $\min _{ E,A }{ { \lambda }_{ 1 }{ \left\| E \right\| }_{ 1 }+{ { \lambda }_{ 2 }\left\| A \right\| }_{ * }+{ \left\| D-ME-A \right\| }_{ 2 }^{ 2 } } $ the M is a ...
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13 views

Complex Least Squares With Magnitude Equality Constraints

For $\mathbf{x} \in \mathbb{C}^N$, I'd like to solve the following problem: $$ \mathbf{x}^\ast = \arg \min_{\mathbf{x}} \Vert \mathbf{Ax-b} \Vert_2 \,\,\,\,\,\, \mathrm{s.t.} \,\,\,\,\, \Vert x_i ...
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35 views

Proving function is convex argmin

How can I show that the following function is convex in which Z is a random variable? $$\rho(Z)=\frac{2}{3}argmin_{t}\{t+10\mathbf{E}[Z-t]_{+}\}+\frac{1}{3}argmin_{t}\{t+5\mathbf{E}[Z-t]_{+}\}$$ I ...
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1answer
71 views

Method to find the extremal values of $xyz$ subject to $x^2+2y^2+3z^2=a$

This question has been asked before but I want to lay out my method and get feedback on reasoning and process this took me a long to put together as I am new to the formatting: Let the function $f$ ...
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1answer
11 views

Build a 4-regular, vertex-transitive, least diameter graph with v vertices

How to build a 4-regular, vertex-transitive, 'least diameter' graph with $v$ vertices? This implies to know what is the minimum diameter of a 4-regular vertex-transitive graph with $v$ vertices. ...
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13 views

Convex solver returns disordered dual variables, how to re-order?

I have the following convex optimization problem: $$ \begin{align*} \min_{x} &f(x) \\ \text{subject to} \\ x_{k+1} &= x_k+\cdots\qquad k=0,1,2,\ldots,N-1 \end{align*} $$ I managed to solve ...
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15 views

Total support in matlab

I need help writing an algorithm in Matlab telling me if a radom matrix has total support or not. I'm trying to use the Linprog formula, but I don't understand it.
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5 views

Limitations of Gaussian Response Surface Methodology for Optimization

I recently went through some material to learn about Gaussian Response Surface Methodology in the field of optimisation. However, I couldn't find the limitations or applications where gaussian ...
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1answer
27 views

Minimize or maximize the powers

I came up with this problem and I could not find a proof. Basically the problem is, suppose positive numbers $a_i$, $i=1,2,\ldots,N$ satisfy $$\sum_{i=1}^Na_i=1$$ then for $p>0$ when the expression ...
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8 views

Can 'Frobenius product method' be used to get analytic expression for **vector derivative**?

this objective function is shown as follow: $$\min_{u*, i*}\sum_{ui}c_{ui}(p_{ui}-x_{u}^Ty_i)^2 + \lambda(\sum_u\|x_u\|^2 + \sum_i\|y_i\|^2) + \lambda_f(\|x_u-\frac{1}{|N(u)|}\sum_{f \in ...
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11 views

Discrete Approximation to Dynamic Lagrangians

Suppose I have the following dynamic optimization problem, where I want to maximize the function $u(c,h)$ over time that's differentiable in both $c$ and $h$. I'm going to assume that the function $u$ ...
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64 views

Why can't this be done? Or can it?

I was writing an answer to this question here From AM-HM $$\frac1{1+x}+\frac1{1+y}+\frac1{1+z}=2$$ $$\frac{1+x+1+y+1+z}{3}\ge \frac{3}{\frac1{1+x}+\frac1{1+y}+\frac1{1+z}}$$ $$\implies ...
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7 views

the optimal solution of $min$ $-x-y+(1/2)(x+2y-3)$

the optimal solution of $min$ $-x-y+(1/2)(x+2y-3)$ s.t. $x,y=0,1,2, or $ $ 3$ Attempt: if we tale the gradient of the objective function we have $[-1/2,0]^T$. This means that y could take any ...
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26 views

Optimize over measure on function space

I'm an absolute newbie in analysis, so this might be a dumb question. Let $S$ the space of non-negative, monotone functions from R to R. Is the following optimization problem well-defined? ...
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1answer
33 views

Relationship between two-equation constrained optimization and one-equation version

I am learning about the Lagrange multiplier. Here's what I understand so far. Suppose a point $P$ is a minimizer of $f(x)$ subject to $g(x)=0$. Then any movement along that level-curve of $g$ must ...
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45 views

Minimize a particular function in one variable

For given $a,b$, what is the minimum value of the following expression? $$ \frac{a}{x^2+b}+x,\qquad x>=0, a>0,b>0 $$ Differentiating the above gives a messy polynomial. I tried plugging ...
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1answer
39 views

How to get explicit solution of the fractional minimization problem?

I need to minimize $f(x) = \frac{x^tQx}{a^tx-b}$, $Q$ is positive definite, $a,b$ are constant vector. But when I take gradient and set it equal to $0$, it's hard to get an explicit expression for ...
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1answer
17 views

Python - CVXOPT: What exactly should I check for G when "Rank(A) < p or Rank([G; A]) < n” exception is thrown?

I am new to using the CVXOPT module for Python and would definitely appreciate any illumination as to why the exception is thrown for my problem. (Also my first time posting a problem anywhere, so ...
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53 views

Probable mistake in calculation of maxima

QUESTION: Given function is $$E=\frac{1}{4}\cdot \frac{F^2}{m}\cdot \frac{\omega_0^2+\omega^2}{(\omega_0^2-\omega^2)^2+4\alpha^2\omega^2}$$ We have to maximise $E$ with respect to $\omega$. MY ...
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17 views

Minimize and maximize the sum of dot products at the same time

this is the problem. I have a set of numerical positive vectors of equal length. For each pair of vectors $(\mathrm{i}, \mathrm{j})$ I define the vector $\mathrm{ij}=\mathrm{i} - \mathrm{j}$. I also ...
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2answers
31 views

Why this two problems are equivalent?

I was reading about Support Vector Machines and I found that it's equivalent to solve the problem of maximize this number: $\frac{1}{\left \| w \right \|}$ with to minimize this number: ...
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1answer
32 views

Matrix-free conjugate gradient

In the conjugate gradient method for solving $Ax = b$, to update the search direction $p$ you would need to evaluate the matrix-vector product $Ap$, i.e. making sure that each search direction are ...
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17 views

Why the dual and the canonical dual may have the same optimal solution?

For instance let be \begin{cases} \max & 3x_1 &+2x_2 &+4x_3 & -x_4\\ &x_1 &+x_2 &-2x_3&&\le4\\ &2x_1&+3x_3&&-4x_4&\ge 5\\ ...
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38 views

Maximise the function with constraints

Is it possible to maximise this function algebraically $$f(x_{1},x_{2})=5\cdot \min\left(\frac{x_{1}}{6},\frac{x_{2}}{2}\right) + 2\cdot\min\left(\frac{1200-x_{1}}{3},\frac{300-x_{2}}{2}\right)$$ ...
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1answer
20 views

Has $\max\{cx:Ax\le b\}$ an optimal solution $x_1=\sqrt 2$ with $A\in \{-1,0,1\}^{m*n}$ with exactly one $1$ and one $-1$?

Let be $A\in \{-1,0,1\}^{m*n}$ with exactly one $1$ and one $-1$ and zeroes at each line. $c\in\mathbb{Z}^n$ such that $\sum\limits_{j=1}^{j=n}c_j=0$. $b\in\mathbb{Z}^m$ positive. How to start to ...
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1answer
14 views

Tree-width of a quadratic pseudo-Boolean function

A pseudo-Boolean function $f : \mathbb{B}^n \mapsto \mathbb{R}$ is of the following form. $$ f \left(x_1, \ldots, x_n\right) = \sum_{S\subseteq V} c_S \prod_{j \in S} x_j $$ Here $c_S \in ...
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51 views

If $x^2+ax-3x-(a+2)=0\;,$ Then $ \min\left(\frac{a^2+1}{a^2+2}\right)$

If $x^2+ax-3x-(a+2)=0\;,$ Then $\displaystyle \min\left(\frac{a^2+1}{a^2+2}\right)$ $\bf{My\; Try::}$ Given $x^2+ax-3x-(a+2)=0\Leftrightarrow ax-a = -(x^2-3x-2)$ So we get ...
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1answer
36 views

A Question On Triple Integration

Can anyone construct a nonzero continuous function $f:[0, 1]\times[0, 1]\times [0, 1]\rightarrow [0, \infty)$ such that \begin{equation*} \int_{t_1=0}^1 \int_{t_2=0}^1 \int_{t_3=0}^1 f(t_1, t_2, ...
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Minimal perimeter of triangle [duplicate]

Given a triangle $ABC$. How one can construct a triangle $DEF$ as $D\in AB$, $E\in BC$, $F\in CA$ and the perimeter of $DEF$ is as short as possible. I found on the net that in acute case the answer ...
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14 views

subdifferential of $\max_{i=1,\cdots,k} x_i+\frac{1}{2}\|x\|_2^2,\ \ \ x\in \mathbb{R}^n$

How to find the subdifferential of $$f(x) = \max_{i=1,\cdots,k} x_i+\frac{1}{2}\|x\|_2^2,\ \ \ x\in \mathbb{R}^n$$ My derivation is: $\nabla \frac{1}{2}\|x\|_2^2=\nabla \frac{1}{2}x^Tx=x$ ...
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24 views

Find the volume of the largest parallelpiped with faces parallel to coordinate planes $x= 0$,$y = 0$,$z=0$

Question : Find the volume of the largest parallelpiped with faces parallel to coordinate planes $x= 0$,$y = 0$,$z=0$ that can be inscribed in one octant of ellipsoid. I tried making some initial ...
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28 views

Is this Feasibility problem NP-hard?

I am trying to solve a combinatorial optimization problem (a feasibility problem) and I have very little idea of solving such problems. The problem is as follows: Solve for $\phi$; \begin{equation} ...
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8 views

Why $x_B=\tilde b +\tilde A x_{\bar B};c^Tx=\psi+\tilde c^Tx_{\bar B}$ doesn't describe an optimal solution iif $\tilde c_i\le 0,\forall i$

How to counterprove the assertion that if a feasible dictionnary in the type \begin{cases} x_B=\tilde b +\tilde A x_{\bar B}\\c^Tx=\psi+\tilde c^Tx_{\bar B} \end{cases} describe an ...
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2answers
60 views

Maximum value of the sum of absolute values of cubic polynomial coefficients $a,b,c,d$

If $p(x) = ax^3+bx^2+cx+d$ and $|p(x)|\leq 1\forall |x|\leq 1$, what is the $\max$ value of $|a|+|b|+|c|+|d|$? My try: Put $x=0$, we get $p(0)=d$, Similarly put $x=1$, we get $p(1)=a+b+c+d$, ...
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1answer
13 views

How to shift points optimally for best rounding

I have sets of points. E.g.: 5.664, 2.292, 1.368, 0.18, 3.3, 4.74, 7.812, 6.564, 5.352, 4.008, 2.568, 5.352 I'd like to shift them a bit (add some uniform dx to all of them) to make them closer to ...
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12 views

Infinite Horizon Transversality Condition

I am an economics student, and I have run into a question where I must apply a transversality condition in order to prove that we have a balanced growth path (all variables grow at the same constant ...
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2answers
72 views

Absolute value and quadratic programming

I would like to solve the following optimization problem using a quadratic programming solver $$\begin{array}{ll} \text{minimize} & \dfrac{1}{2} x^T Q x + f^T x\\ \text{subject to} & ...
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14 views

Finding a function of a random variable that maximizes some expression

The following problem is part of my studies, so I would prefer hints or suggestions for self-study. Let $v_1$ be a random variable taking values in $[a,b]$ for $a,b\in \mathbb R$ and assume that the ...
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1answer
26 views

Minimum of a bivariate quadratic function

According to (hope my calculation below is correct) https://en.wikipedia.org/wiki/Quadratic_function a bivariate quadratic function is a second-degree polynomial of the form $$ ...
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13 views

optimization problem involving matrices

This optimization problem is confusing me. Assume you are looking for the best matrix ${\bf X}$ and you have a matrix ${\bf V}$. I have the following two optimization problems $${\bf X}^*= ...
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18 views

Optimization under uncertainty, solving for optimal value of $v_1$

I want to solve the following function by finding the optimal value for $v_1$. $$\max_{\begin{array}{c}v_1,\beta_1 \\ 0<\beta_1<1 \\ v_1>0\end{array}} ...
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1answer
56 views

Maximum of function containing two variables $x$ and $y$

If $x+y+\sqrt{2x^2+2xy+3y^2} = k(\bf{Const.})\;,$ Then $\max(x^2y)\;,$ Where $x,y\geq 0$ $\bf{My\; Try::}$ Let $x^2y=z\;$ Then we get $$x+\frac{x^2}{z}+\sqrt{2x^2+\frac{2z}{x}+\frac{3z^2}{x^4}} = ...
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1answer
31 views

Show that a differentiable function $f:\mathbb{R} \to \mathbb{R}$ has a global max in $a$ if $a$ is its local max

My task is this: Let $f:\mathbb{R} \to \mathbb{R}$ be a differentiable function and assume that the only stationary point $f$ has is a local max in the point $A = (a,f(a))$. Show that $A$ must be a ...
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53 views

Optimization with a Probability

Imagine two points in $ℝ^2$ at $(-1, 0)$ and $(1, 0)$. You would like to walk from one point to the next in the shortest distance possible. However, there is a line segment coming from the origin to a ...
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1answer
37 views

piecewise linear minimization equivalent to linear programming

Why is \begin{equation} \begin{aligned} & \min\max_{i=1,\ldots,n} & &a_i^Tx+b_i\\ \end{aligned} \end{equation} equivalent to an LP \begin{equation} \begin{aligned} & \min & ...
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36 views

Maximum flow on a directed, acyclic graph

What would be the best algorithm to use for finding max-flow/min-cut on a directed, acyclic graph with integer flows, capacities, and vertex demands? I've been thinking Dinic's Algorithm would be ...
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0answers
24 views

How to solve exponential matrix factorization with constrain: $UV^T>0$

recently I would like to optimize the following loss function: $$L=\sum_{ij}W_{ij}(X_{ij}-exp(-\sum_{l} U_{il}V_{jl}))^2$$ $$s.t. \sum_lU_{il}V_{jl} > 0$$ Where $W \in \mathbb{R}^{m \times n}, X ...
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1answer
16 views

Solving a polynomial equation along a set of lines numerically.

Assume that I for some reason want to solve multidimensional polynomial equations $$p(x_1,x_2,\cdots,x_k) = 0$$ or possibly (if there is no solution) $$\min_{\forall x_{.}} \{p(x_1,x_2,\cdots,x_k)\}$$ ...
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1answer
41 views

Books on Statistics and Optimization

I'm trying to close gaps in my education especially in Statistics and Optimization theory. I had an awful class on Statistics so I want to learn it by myself. As for Optimization we had a pretty good ...
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23 views

Probability of an event occuring $n$ times, given that it can occur $n$ times or does not occur at all.

Suppose you have an event whose probability is $\rho$. This event either does not occur at all or occurs $n$ times, because when it occurs once, all the others occurrences are linked to the first. ...