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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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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0answers
16 views

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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0answers
15 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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0answers
25 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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0answers
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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0answers
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
62 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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0answers
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 ...
3
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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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0answers
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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0answers
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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0answers
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
38 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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0answers
37 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. ...
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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
15 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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15 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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0answers
182 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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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 ...
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1answer
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Periodic point of dynamical system

Hi please help me someone with the proof: We have a function $f:\mathbb{R}\longrightarrow\mathbb{R}$ continous and invertible, discrete dynamical system is given by $x_{n+1}=f(x_n)$ (a): prove that ...
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0answers
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Find $\alpha, \beta$ s.t. the following is minimized

Hello I would like to find $\alpha,\beta$ s.t. $$ e(\alpha,\beta) = ||\sqrt{1+\gamma^2}-\alpha-\beta\gamma||_\infty = || f_{\alpha,\beta}||_{\infty} $$ Is minimum (consider $\gamma \in [0,1)$, ...
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2answers
39 views

Optimization of a function over probability distributions

I'm trying to solve certain optimization problems dealing with probability distributions. Consider the space of probability distributions $\{ 1, ..., N\} \to [0, 1]$ I have a function $f : (\{ 1, ...
2
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0answers
51 views

How to use Expectation Maximization (EM) in Item Response Theory (IRT)?

Could you give a worked example on the steps of Expectation Maximization in Item Response Theory if we use the Two Parameter Rasch Model. The student abilities are unknown and the question parameters ...
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Modeling simple linear equations

This should be pretty simple but I'm blanking on this. I need to model (graph) how path 1 becomes equally as efficient as path 2 as the distance of path 2 increases. distance of path 1 (from A to B) ...
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14answers
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Why does a distance and its square reach their minimum at the same point?

There is a question in my calculus textbook that asks to find a point on the parabola $y^2 = 2x$ that is closest to point $(1,4)$. They want us to first use the distance formula, but then proceeded ...
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1answer
23 views

Constrained maximization of …

I have to maximize $U(x,y)= Min(ax+y, by+x)$ s.a $p_{1}x +p_{2}y =m$. I try the traditional solution for a leontieff $(ax_{1}+y= by_{1}+x)$ function but I'm not sure.. beacause exist regions where one ...
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23 views

analytical solution to solve maximization problem

My question is the following: Let $f(x_1, ..., x_n, \theta): \mathbb R^{n+1} \rightarrow \mathbb R$ be a function in which we are interested in maximizing in terms of $\theta$. The traditional ...
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optimization problem from my textbook

Given the objective function of a constrained optimization problem is $f(x₁, x₂)= c $ and the constraint is $g(x₁, x₂) = b$. How can I Show with a diagram that a unique optimum solution exist; unique ...
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+50

Are any tools or techniques available to solve the “placement of safety points” problem?

Definition 0. Given a metric space $X$ and subsets $H$ and $S$ thereof, define: $$d(H,S) = \sup_{h \in H} \inf_{s \in S}d(h,s)$$ (This as an asymmetric version of the Hausdorff distance.) Here's ...
1
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0answers
16 views

Variational Calculus with Discrete Objective

I'm trying to infer a smooth, non-negative function from some given data ($\vec{m},\vec{\alpha},\vec{\beta}$). That is, I want to solve (I think) $$ \mathop{\arg\!\min}_{g \in ...
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1answer
22 views

A Basic Question E-views

I ask a question about E-views. Is the P-value in the picture less than 0.05 or greater than 0.05? I'm confused because of the presence of the sign '<' in front of 0.10. Please help mee. Thank you. ...
4
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2answers
48 views

L1 regularized unconstrained optimization problem

I am encountering an unconstrained minimization problem. The problem is of the form $$\min_x \frac{\|x-a\|_2^2}{2}+\lambda\|x\|_1$$ where $x,a \in R^n$ and $x$ is the optimization variable. ...