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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How do I set a lower bound to the solution's norm in a QP problem

I know that LASSO-regularization can be used to scale into an $L_1$ upper bound for a solution. But what if I want the norm to be within a specific range $[a,b]$? ie. I also want to set a lower bound? ...
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Convexity of matrix inverse

If $$f\colon\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}$$ where $$f(x,y)=y^T x^{-1}y$$ and dom$(f)=\{(x,y)\ |\ x+x^T\gt 0\}$, then is $f$ convex?
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The area visible from two lighthouses with angle of vision 30 degrees, built at distance 10km from each other

The distance between 2 lighthouses is 10 km. What is the maximum area of the ocean in which both can be simultaneously visible if the angle of vision for each lighthouse is 30 degrees?But the minimum? ...
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51 views

Infimum of $\frac{||u'||^p_{L^p}}{||u||^p_{L^p}}$ for $u \in W^{1,p}_0((0,1))$

Good afternoon everyone! It is very easy to show that the infimum mentioned in the title is strictly positive, but it seems much more difficult to show that it is attained within the Sobolev space of ...
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refomulation of an optimization problem

I have written a program for optimizing a set of generators. And I need to reformulate this problem, to include additional generators and constraints. I have hourly price and cost data and need to ...
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36 views

Maximal Multiplication of All Possible Summands

I have recently got interested in the following problem: Give a decomposition of a natural number to natural summands whose multiplication is maximal. I have tried to solve this problem, and ...
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63 views

How to minimize the number of functions to be projected on

I have a set of functions $f_i,\, i=1,2,\ldots,n$ defined on an interval $[a,b]$, and a function $F$ also defined on $[a,b]$. I would like to project F on a subset of functions $f_i$ so that the ...
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Computing $\sup_{\left \| u \right \|=1} d(u,F)$ in a closed subspace of a normed space.

I have come across with the following problem: Let $E$ be a normed space and $F$ a closed subspace of $E$. It's asking to compute $\sup\limits_{\left \| u \right \|=1} d(u,F)$. What I know it's ...
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23 views

Convexity over a line given a convex interval [duplicate]

Let $f : \mathbb{R}^n \to \mathbb{R}_∞$ be a function. I want to prove that $f$ is convex over the line $L_{v,x_0}$ iff $\psi : \mathbb{R} \to \mathbb{R}_∞$ $\psi(t) := f (x_0 + tv)$, is convex ...
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Can gradient descent solve this problem $\text{argmin}_x \|Ax-[Var(Ax)]^{\frac{1}{2}}-b\|^2$?

How can I find the (approximate) solution to the following problem: $$\text{argmin}_x \|Ax-[Var(Ax)]^{\frac{1}{2}}-b\|^2,$$ where $Var(.)$ denotes the variance? $A$ is matrix and $b$ and $x$ are ...
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How to find the extrema of a function?

I'm having problems finding the extrema of the function $h(x, y) = 2x sin(y) + y^2−x^2$. There is supposed to be one saddle point but I can't seem to get that. I tried taking $f_{xx}f_{yy} - ...
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29 views

incremental approach to solve positive least square problem

Is there any incremental (approximate) solution for the following positive least squares problem: $$\min_x \|Ax-b\|^2\qquad \textrm{s.t.}\qquad x_i> 0,~b_1=1,~b_{i>1}=0$$
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Two quadratic programming problems always same answer?

Was exploring quadratic programming optimization and for two types of problems the answers seemed to always equal. Is there an intuitive proof? Problem 1: Minimize $\tfrac{1}{2} \mathbf{x}^T ...
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28 views

How do you calculate Up and Down Penalties on a Branch and Bound algorithm of a MILP?

My notes really don't explain this clearly at all, so I have no idea what to do. If I have the following MILP: In which I've been told to solve it using: (a) Rule 1 (choose the variable with the ...
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17 views

Optimal design for constrained Bayesian slope intercept model

Here is a problem I've been stuck on for quite a while. Consider the model \begin{equation} \mathbf{y}=\mathbf{H}\pmb{ \theta }+\pmb{\epsilon }. \end{equation} The design matrix is given by: ...
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95 views

Implicit function theorem in comparative static problem

The individual lives for two periods. He has a utility function $u(c_{1} )+ bu(c_2)$. His budget constraint requires that his period I consumption be his period I endowment minus any savings, $c_1 = ...
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Solving linear functions with constraints.

I have a table of data, for maximum available items in a given time period. With a constraint on how many of each items I can take in total. When I have taken the allowed amount of an item I can ...
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250 views

Optimization: maximum area of a triangle under a parabola

Optimization: maximum area of a triangle in a parabola Inside a curve ($x^2-25$ - Parabola) a triangle is drawn with A as the vertex at the origin and the line joining points B and C lie on the ...
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25 views

Unique maximizer

I have two functions, $f(x,y)$ and $g(x,y)$. $x\in[0,X]$ while $y\in [0,Y]$. Both functions are non-negative on the domain. Further, $f$ is increasing in $x$ and decreasing in $y$. $g$ is the ...
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Combinatorial Optimization and Relaxation

There are a number of NP-hard optimization problems that may be formulated as either binary linear or quadratic programs, i.e. $\min_x c^tx $ s.t. $x \in K, x_i \in \{0,1\}$ or $\min_x x^t Q x $ s.t. ...
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Sign of the Lagrange multiplier associated with an equality constraint

I am trying to determine conditions under which the Lagrange multiplier(s) associated with an equality constraint is(are) positive. In general, the multiplier of an equality constraint is not ...
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31 views

Maximization question [duplicate]

I'm trying to find the maximum value of the function $f(x,y)=(ax+by)^p+x^p$ subject to the constraint $x^p+y^p=1$. Here, $a,b$ and $p$ are constants with $a,b>0$ and $p>1$, and $x,y>0$. I ...
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Max - min problem of a quotient of norms

For the $2\times2$ matrix $\begin{bmatrix}4&0\\-3&-5\end{bmatrix}$ Part 1 Find nonzero vectors $u$ and $w$ that maximize and minimize respectively the quotient $||Av|| / ||v||$. Part 2 ...
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How to solve sum of sines and cosines system of equations?

I have a set of equations to solve which in the following form: $ \cos(t_1 + t_2 + t_3 + t_4) + \sin(t_1 + t_2 - t_3 + t_4) + \cos(t_1 - t_4 + t_3 - t_5) + \sin(t_1 - t_2 + t_3 - t_5) + \cos(t_1 + ...
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How to interpolate a function with a reproducing kernel

I am trying to interpolate a function that is noisy, but I know with a high amount of certainty about a third of the points in the series. I am trying to estimate the smooth mean of the signal via a ...
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29 views

Does point's neighborhood have no local extremum?

I have polynomial of some limited degree: $f(x,y) = a_1 + a_2x + a_3y + a_4xy + a_5x^2 + a_6y^2 + a_7x^2y + \ldots$ There is a point $p_0=(x_0,y_0)$, which is NOT a local extreme NOR inflection ...
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65 views

When is $D \max G = \max D G$?

All matrices are real. The operator $\max$ on matrices returns the largest value in each row. We are interested in characterizing the set of matrices $D$ of size $n \times m$, $m < n$ such that we ...
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113 views

A maximization problem

I'm trying to find the maximum value of the function $f(x,y)=(ax+by)^p+x^p$ subject to the constraint $x^p+y^p=1$. Here, $a,b$ and $p$ are constants with $a,b>0$ and $p>1$, and $x,y>0$. I ...
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Maximizing expected value when distribution is binomial

Consider the following problem: $$\max_{n\in\mathbb N}\;f(n)= \frac12 \left[v_0 \sum_{i=\lceil k_n \rceil}^n \binom{n}{i}p^i (1-p)^{n-i} + v_1\sum_{i=1}^{\lfloor k_n ...
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Trouble with formulation of objective function (constraint optimization)

I am new to optimization and I will try to state my question as clear as I can. I need to solve this constraint optimization problem. I want to find real vectors $\mathbf{f}$ and $\mathbf{g}$ that ...
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SOS relaxations for polynomial optimization

I do not understand how SOS (Sum-Of-Squares) relaxation for polynomial optimization works in some cases. For instance, consider the polynomial optimization problem: \begin{equation} ...
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How to understand ' Let $\mathcal{H}$ be a Hilbert space of functions $f$ : $ \mathcal{X} \rightarrow R$, denoted on a non-empty set $\mathcal{X}$.'

I am a beginner. By asking this question, I means that, to construct a Hilbert space, should $\mathcal{X}$ satisfy some properties? Furthermore, in some papers especially on machine learning, ...
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125 views

Dynamic programming problem for discrete linear time varying system

I'm working on a linear time varying discrete(LTV) multi input multi output(mimo) system. I formulate the problem description in the following way $$x_i(k+1) = x_i(k)\cdot A_i(k) + B_i(k)\cdot ...
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Maximising the Area of a Cyclic Quadrilateral

In cyclic quadrilateral $ABCD$, $AB = AD$. If $AC = 6$ and $AB/BD = 3/5$, find the maximum possible value of $[ABCD]$. (Source: SMT 2014) If we let $AB=AD = 3x$ and $BD=5x$, from Ptolemy, we have ...
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The optimization problem of soft margin Support Vector Machine: How to interpret?

I try to understand what exactly we are trying to optimize in the case of Support Vector Machine problem, which supports soft margins. The original problem is posed first as, without soft margins ...
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Show that weak local minimum of a convex function $\mathbb{R}^N\rightarrow \mathbb{R}$ is its weak global minimum.

Show that weak local minimum of a convex function $\mathbb{R}^N\rightarrow \mathbb{R}$ is its weak global minimum. Does the same happen to strong minimums? I know that when $f$ is convex, then we ...
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39 views

What is the best choice given a probability and a cost for each choice?

I've been dealing with this problems for a few hours now and think I could use some outside help. The scenario is the following: We are given different choices with each one having a probability of ...
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Why is this conic dual problem infeasible?

The problem is: $$\min \ x_2 : Ax -b = [x_1 \ 2x_2 \ x_1]^T \ge_{L^3} 0$$ where $L^m$ is the Lorentz cone. Which I found to have an optimal solution when $x_2 = 0$. I have shown that the conic ...
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39 views

Generating primal solution from dual solution of a LP

How to get the primal solution from a dual solution in general? For example, let the primal problem is $$ \text {maximize } 2r_1+2r_2-2c_1-2c_2 $$ where $$ r_1-c_1\leq1\\ r_1-c_2\leq1\\ ...
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Notion of complex optima

Consider the function: $$y = \frac{1}{3}x^3 + x$$ Suppose we wanted to determine its local optima, but instead of looking at local optima with domain $R$ we instead consider domain $C$ and range ...
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KKT for not convex problems

In my optimization course we learned something about KKT for not konvex problems: $$min \; f(x)$$ $$s.t. \; c(x)=0$$ $$d(x)\geq 0$$ $$f(x): \mathbb{R}^n\rightarrow \mathbb{R}$$ $$c(x): ...
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Optimal value of decision variable leads to inconsistency

$\epsilon$ is a random variable with support in $(0.8,0.95)$ and pdf $f(\epsilon)$. The following equation arises out of a business problem: $ENP=800*A*E(\epsilon)+ 9000 - ...
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Minimization using logarithmic barrier function

I'm thinking of the quadratic problem(QP) \begin{align} &\underset{x\in \mathrm{R}^n}{\mathrm{Minimize}}\ \ \ \frac{1}{2}x^\top{}Qx + f^\top{}x\\ &\mathrm{subject\ to}\ \ \ \ a_ix \leq b_i\ ...
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What is $F_P$ and $E(P)$?

I'm reading Handbook of Graph Theory: At this section, he speaks about $F_P$ and $E(P)$. It's not really clear what they are. I guess there is enough context for someone to answer me but if ...
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48 views

Maximum / Minimum Cost of a Box

this is a sample final question for a multivariable calculus course. "A rectangular box has two opposing sides (left and right) made of gold, two (front and back) of silver, and two (top and bottom) ...
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Optimization of a function

I need to optimize $$f(x,y,z)= x^2-y+e^{z}$$ with the restriction $$(x-2)^2+(y-3)^2+z^2=1$$ I've tried to substitute the restriction in $f(x,y,z)$ but it seems not to work. And when trying to use the ...
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Selection of the mean of random variables to optimize the expected value of objective function

Here is the objective function to be maximized: $$ E_{v}(\log(1+v^{\mathsf T} \Lambda v) ) $$ where $v$ is a Gaussian distributed random variable vector $v ∼ \mathrm{CN}(M,I)$ with its mean vector ...
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Find the Maximum and Minimum of the Given Function on the Given Plane Region

I've been good with most of the max/min finding in different regions, but this one's really messing with me. Can anyone lend a hand? Thanks. z = 2xy Region is the circular disk $x^2 + y^2 =< 1 $
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Maximize area of a corral

See problem 7 and figure 9 in picture What I've done so far: Not sure if $P=2l+2w$ or just $l+2w$ (dashed line makes me think the latter) $600=\pi r+l+2w$ $600=\pi r+2r+2w$ $w=\frac{600-\pi ...
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Derivative of a trace w.r.t matrix within log of matrix sums

I'm trying to solve an optimization (sub)problem and am running into trouble with a tricky derivative. I'd like to find the matrix $C \in \mathbb{R}^{n\times d}_+$ which minimizes the following ...