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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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
32 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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2answers
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
16 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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3answers
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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1answer
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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0answers
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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4answers
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
24 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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2answers
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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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. ...
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1answer
18 views

Elementary derivation of max/min of quadratic trig polynomial

Let $\alpha, \beta, \gamma, \delta$ be fixed real numbers, and $x$ a variable in $[0,\pi)$. Consider the expression \begin{equation} (\alpha^2+\beta^2)\cos^2(x) + ...
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1answer
14 views

Is it correct to write $argmin(x, y) \sum_i^n |p_{x_i} - x| + |p_{y_i} - y| = argmin(x) \sum_i^n |p_{x_i} - x| + argmin(y) \sum_i^n |p_{y_i} - y| $?

$argmin(x, y) \sum_i^n |p_{x_i} - x| + |p_{y_i} - y| = argmin(x) \sum_i^n |p_{x_i} - x| + argmin(y) \sum_i^n |p_{y_i} - y| $ Is it a legit way of separating argmins to show independence of $x$ and ...
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3answers
27 views

Understanding when to use the chain rule when differentiating trig functions.

I'm trying to solve an optimization problem that involves finding the maximum angle that subtends two points. The two points are $b = (0, 5)$ and $t = (0, 14)$. The third point is the point that is ...
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1answer
20 views

Method for calculating minimum number of transmissions?

(This is a real issue I face.) I have $42$ files I want to transmit. I tried sending them in a single archive but four of them had issues, and as a result the entire archive was rejected. I do know ...
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0answers
14 views

Can the time complexity of maximum-flow algorithm using fattest path method be represented by |V| and |E| only?

I've got a problem with "fattest path" heuristic in Max-Flow algorithms. ( http://www.eecs.berkeley.edu/~luca/cs261/lecture10.pdf ) The problem is 'prove or disprove that the time complexity can be ...
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1answer
63 views

Show f takes on maximum boundary for function

Suppose $\Omega$ is a bound set in $\mathbb{R}^2$ and $\bar\Omega$ its closure. Assume $f\in C^2(\Omega)\cap C^0(\bar\Omega)$. Moreover, assume $f$ satisfies the partial differential ...
6
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177 views

Construct a matrix of polynomials to optimize condition-like score

I'm a physicist currently working on my PhD. Within my studies, my colleagues & I encountered a (strictly mathematical) problem that baffles us (and anyone else we've talked to so far) and is also ...
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0answers
15 views

Basic Linear Algebra/Root finding question

What is the general method for solving this problem? $\theta_n.1_T'.z_T=0_n$ where $\theta_n$ is a n x 1 vector of parameters that are free to vary, $1_T'$ is a 1 x T vector of ones, $z_T$ is a T x ...
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0answers
10 views

Optimizations in Laplacian Eigenmap/Graph Embedding?

Note -- this question is closely related to this question that asks why the optimization constraint has to be $y^TDy=1$ instead of simpler $y^Ty=1$. Maybe answering this question will automatically ...