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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Using optimization for a logarithmic function

Question: A tangent line is drawn on the graph of $y=\ln x$ for $0\lt x\lt 1$. A right triangle is thus formed in the fourth quadrant. If we regard the area of this triangle has a positive value, ...
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How can we constrain lagrange multipliers in svm dual by adding constraints in primal problem?

Consider svm-dual,i.e., \begin{align} &\text{maximize} \sum_{i=1}^n \alpha_i-\frac{1}{2\lambda} \sum_{i,j=1}^n \alpha_i \alpha_j y_i y_j K(x_i,x_j)\cr &\text{subject to, } 0\leq \alpha_i ...
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Am I headed in the right direction with this area optimization question?

Question: A fence will create a rectangular area with one side being formed by an existing building (and hence, the fence only needs 3 sides). One side will be created using Redwood fencing and the ...
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Is this a Combinatorial Optimization problem with Multiple Constraint Satisfaction?

Given n-dimensional data consisting of over 20000 samples with 200 dimensions, using this as an example: ...
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1answer
68 views

Knapsack in graph

This question is from job interview for a software company.   "You are given an undirected connected weighted graph with $n$ nodes. The weight function represents transportation costs. In ...
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37 views

Maximum number of teams of three people such that each team is built in one of two ways

A coach picks team members in two ways:   A. The team of three people should consist of one experienced participant and two newbies. Thus, each experienced participant can share the ...
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1answer
12 views

Multi objective optimization: Ideal vector

I'm going to consider the two problem distinctly. Now I want to calculate $z_1^{id}$ and $z_2^{id}$ and $x_1^{id}$ and $x_2^{id}$ where $z_1^{id} = min(x)$ $z_2^{id} = max(y)$ $z_1^{id}$ is the ...
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How to transform this problem to a matrix optimization problem?

I wonder how can I loose the following set of equations to an optimization problem ? Suppose given three real vectors $w_0$, $f$ and $\delta$, and a positive entry vector $c$, such that for every ...
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Why lower semicontinuity?

I'm reading a proof on the existence of a solution to a minimisation problem, but I'm stuck. I give a brief summary of the arguments up to the point at which I'm stuck(at the yellow box). ...
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Auction Design : Multiple lots, one win max per bidder, not regret

This is a real life game theory problem. I have to organize an auction. There is a finite number of lots, which are not equivalent. There is a finite number of bidders; the number of bidders is ...
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1answer
55 views

Area of the figure within the circle and outside a polygon

For which values of the parameter $c \in \mathbb{R}$, the area $S$ of the figure $F$, consisting of the points $(x,y)$ such that $$\begin{gathered} \max \{ \left| x \right|,y\} \geqslant 2c \hfill ...
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gradient flow on $SU(n)$

Define the following cost functions $f_1, f_2 :SU(n) \rightarrow \mathbb{R}$ by $f_1(U) = Re \left( \text{Tr}\left(G^{\dagger} U \right) \right)$ and $f_2(U) = \left| \left( \text{Tr}\left(G^{\dagger} ...
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1answer
40 views

An optimization problem, in the form of a word problem,

The manager of a $1000$ seat concert hall knows from experience that all seats will be occupied if the ticket price is $50$ dollars. A market survey indicates that $10$ additional seats will remain ...
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Find the function $\hat{g}$ that maximizes $\int_0^1 x^2g(x)dx$ over the set of all functions that satisfy the following conditions:

Find the function $\hat{g}$ that maximizes $\int_0^1 x^2g(x)dx$ over the set of all functions that satisfy the following conditions: $\int_0^1 |g(x)|^2dx =1$, $\int_0^1 g(x)dx=0$, and ...
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2answers
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Bivariate optimal density

Consider any feasible $p:[0,1]^2\to [0,1]$ that allows discontinuities and the problem $$\min_{p(.)} \int_0^1\int_0^1 p(x,y)^2 dF(x) dG(y)$$ s.t. $$\int_0^1 p(x,y)dG(y)=k\phantom{0} for \phantom{0} ...
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2answers
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Differential equation for finding closest point on surface.

Inspired by this question I got to think about a more general case. Say I have any discretized surface and want to find closest point from each point outside of surface to the surface. Say that I can ...
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Matrix Factorization with Arbitrary Dimensions

Continuation of a previous question here. Suppose I have a $n\times m$ matrix $A$. I choose some $k$, and want to find a factorization $A=XY$ where $X$ is $n\times k$ and $Y$ is $k\times m$. In ...
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3answers
120 views

How can I find the minimum value for $F(x,y,z,w)=x^2+y^2+z^2+w^2+xy+zw-xz-yw-yz$

Let $x,y,z,w$ be integer numbers,and $xw=yz+1$ Find this minimum of the value $$x^2+y^2+z^2+w^2+xy+zw-xz-yw-yz$$ This is how did it and I would like to know if I made a mistake Let ...
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32 views

Shannon Entropy Maximization with Constraints

I have got a cumulative distribution function $F_X(x)=Pr(X<=x)$. This distribution is described by 2 parameters $\alpha, \beta$. We define $F_k$ as follows: $\forall k<=n_k, ...
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Convexity of a scalar function

I was trying to solve the following problem: $min_A ||AC_iA^T - C_R||_F^2$ Where both $C_i$ and $C_R$ are symmetric matrices. They represent the covariance of RGB colors, and usually it's highly ...
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How to write my objective function?

I am working on an optimization problem. I need help writing the objective function. Therefore, I have the roads modeled in a 2D plane, and found the probability of loss of charge, for an electric ...
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Minimizing $h(f(x,y))$ by choosing $f(x,y)$, given conditions on $f$

I have a function $$h(f(x,y)) := \frac{\int_{1}^2 f(v,v)dv (1-\int_{1}^2 f(v,v)dv))}{- \int_{1}^2 f(v,v) v\ dv\ f_y(2,2)}$$ which I want to minimize by choosing an appropriate function $f$ which ...
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Would like some help formulating an optimization problem

I have a function $f$ that takes $n \geq 1$ positive real-valued arguments $\mathbf{a} \in R^n_+$. This function is defined for all amounts of inputs (e.g. $f(1)$ and $f(3, \pi, 17)$ are both valid) ...
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Optimization problem for maximum volume of inscribed figure. [duplicate]

While studying, I came upon this problem: "What is the largest possible volume a right circular cylinder can have if it is inscribed in a sphere of radius 5?" The answer was shown as ...
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Numerically/Computationally estimating parameters

I have a function $f(x)$ and I have an estimating function $\hat f(a,b,c,d;x)$ Say, I also have a scoring function $S(f,\hat f,x)$ (which could very well be mean square error) And I have some ...
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optimization of a function with inequality constraint

I have a function to be maximized subject to constraints. I can write the primal Lagrange function as the following: (objective function WITH two constraints in the last two terms) $$L_P = ...
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1answer
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Is there any method that convert a concave problem into convex problem?

I have an optimization problem of the form: \begin{align} \begin{cases} x_2 \rightarrow \min, \\ \text{subject to:} \\ f_1(x) \leq 0, \\ f_2(x) \leq 0, \end{cases} \end{align} with $x= (x_1,x_2)^T$ ...
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optimizing over a set of symmetric matrices

I need to minimize a complicated loss function, $f\left(\Lambda\right)$ over a set of symmetric matrices, $S_{p}$ of dimension p, such that all the eigenvalues of $\Lambda \in \left[0,1\right]$. I ...
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Is there a name for this type of online optimization problem?

I have a sequence of items $1\leq i \leq n$ that arrive to me one at a time. Each item has a weight $w_j\geq 0$. If I pick up one item, I will not be allowed to pick up any of the next $k$ items ...
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Reliability/survival function raised to a power

Let $r(p)$ be the reliability function, and suppose that $r(p)=r(p,p,...,p)$ and that $r(p_0)=p_0$ for a certain $p_0$, $0\leq p,p_0\leq 1$. I'm asked to prove that $r(p)\geq p$ if $p\geq p_0$ and ...
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How to find the shift that minimizes the difference between two vectors?

I am looking for a efficient way to find the value of k that minimizes $\sum(s_t - b_{t+k})^2$ where $s$ and $b$ are N-dimensional vectors and the values are wrapped around like this: $b_{t+k} := ...
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multi-objective reduction of a given set

I have a set of arguments $v_k$. Each argument has a set of two different numeric values $x_{ak} \in [0,\infty]$ and $x_{bk} \in [0,\infty]$ associated to it. The set $V$ contains all $v_k$s. I’m now ...
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1answer
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How come $Ax\le b$ and $c^Tx\ge \alpha +\epsilon$ has NO nonnegative solution.

Let $\alpha=c^Tx^*$ be the optimum value of the standard form of (LP)(= max $c^Tx$ subject to $Ax\le b$ and $x\ge0$ in $\mathbf{R^n}$) Then we know: $Ax\le b$ and $c^Tx\ge \alpha$ has a nonnegative ...
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How to optimize the repartition of samples in weighted channels?

This is more like an applied mathematics question, so my apologies if I am at the wrong place. Let S(n) be an infinite sequence of real numbers strictly growing from 0 to 1 (asymptotically). Let P be ...
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Is this function convex or non-convex? How do you decide?

The problem is: find $$\min⁡ \mathrm{P}\left[{\log(1+p||H^H \mathbf{w}||^2)\over 1+p||G^H \mathbf{w}||^2}<R\right]$$ constraint to: $||\mathbf{w}||^2=1$ where $H$ and $G$ are matrices of ...
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Optimize distributions for low mean, high variance

Assume a context with $N$ approximately normal distributions where a lower mean implies a 'better' distribution and a high variance or high standard deviation implies a 'better' distribution as well. ...
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Optimization problems with combinations of a finite set as the feasible area?

For example: Provided that $S\subset \Re$ is a known finite set ($n\leq |S| < \infty$), number $k$ is known, and $1 \leq k<n$ minimize $f(x_{1},\ldots, x_{n}) = \sin (\sum_{1\leq i\leq ...
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projection KKT optimal condition

Using the KKT optimality condition find the orthogonal projection of an arbitrary point $c \in$ to the closed convex set $C$ (non empty) defined by: (a) $C=\{x \in R^n : Ax\leq a\}$ where $A\in ...
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find the minimum value of this integral when $1>t>0$, $f(t)=\int\limits_0^1 x |e^{-x^2} - t|\ \mathrm{d}x = ?$

Is there someone who can show me How do i find the minimum value of this integral when $1>t>0$, \begin{align*}f(t)=\int\limits_0^1 x |e^{-x^2} - t|\ \mathrm{d}x &= \end{align*} Note : ...
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KKT Optimality Conditions

I am working with the following optimization problem: $$ \min_{\Delta} \boldsymbol{\theta}^T\boldsymbol{\Delta} \\ \text{Such that:} ~~~0 \leq \mu_i + \Delta_i \leq 1 ~~\forall~~ i\in\{1,2,\ldots, n\} ...
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1answer
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Campbell's Source coding

In the usual Shannon's source coding problem one chooses code words that minimize $E[L]:=\sum_i p_il_i$ over all $L=(l_1,l_2, \dots), l_i\ge 0$ such that $\sum_i e^{-l_i}\le 1$ (Kraft inequality), ...
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E-olymp: Cake. Giving Wrong Answer

Cake This is a e-olymp programming question mathematical optimization. In honor of the birth of an heir Tutti royal chef has prepared a huge cake, that was put on the table for Three Fat Man. ...
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1answer
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Generating vectors in a non-orthogonal 3D lattice with increasing magnitude

I am trying to build an algorithm to generate a sequence of lattice vectors $\mathbf{v}_n$ in 3D such that: (a) the first vector $|\mathbf{v}_1|$ is the shortest vector of the lattice (b) for all $i ...
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A special case of GUBMKP

I am seeking a problem that resembles the Multidimensional Knapsack Problem with Generalized Upper Bound Constraints where the resources available are of equal sizes.I am only getting the case where ...
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1answer
62 views

Optimization of solve

Find the minimum value and the maximum value of the function $$y(x)=\frac{x^3}{x-3}$$ when $4\le x\le5$ I found that $f(x)$ is decreasing on the interval $[4,\frac{9}{2}]$ and increasing on ...
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1answer
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Image restoration in matlab via PDE toolbox

I want to remove a noise for an image using matlab, when the observed image is $$f=u+v$$ where $u$ is the restored image (is the image i want recovered) and $v$ is the gaussian noise. To restore $u$, ...
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1answer
36 views

Local extrema in special directions

I am looking for the extrema of a function $G(y_1,y_2,y_3,y_4)$ subject to the constraint $y_1 = y_4 + y_2y_3.$ We know that $G$ is defined if $(y_2,y_3,y_4)$ is in the cylinder $\mathbb{D} \times ...
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1answer
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How to resolve total variation $F(u)=f(u)+\lambda\sum_i^N\int_{\Omega}|\nabla u_i(x)|dx$

Given $u(x)=[u_1(x)..u_N(x)]$, $0 \le u_i(x) \le 1, \sum_i^N u_i(x)=1$ and the cost function is: $$F(u)=f(u(x))+\lambda\sum_i^N\int_{\Omega}|\nabla u_i(x)|dx$$ where $u_i(x)$ is a value that indicate ...
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1answer
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Is it covex function?$J_{new}(u)=\int_{\Omega} \sum_{i=1}^{N} \lambda_if(x)u_i(x)dx$

I have a function such as $$J(u)=\int_{\Omega} \sum_{i=1}^{N} f(x)u_i(x)dx$$ where $f(x):\Omega \to R$, $0 \le u_i(x) \le 1,\sum_i u_i(x)=1$ Given that $J(u)$ is a convex function w.r.t $u$. Now I ...
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1answer
48 views

Derivation of energy function

Given the following energy function $E(d)$ (also found here on page 3): $$ E_d = \sum_{x,y \in \Omega} \left(d_{x,y} - \hat{d}_{x,y}\right)^2 + \lambda \sum_{x,y} \left( ...