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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Given matrices $B$ and $C$. What is the value of $L$ that minimizes the value $||L^T \times B \times L - C||_F$?

Where $L \in R^{m \times n}$ and $B \in R^{m \times m}$ and $C \in R^{n \times n}$ $B$ and $C$ are symmetric positive semi-definite. Where $\times$ denotes matrix multiplication and $||.||_F$ ...
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26 views

Minimum of a function given by integral and inequality type constraint

I need your help with the following problem I want to minimize $$2a + \int_0^1 tx(t) \, dt \to \min$$ s.t. $$1 - a - \int_t^1 x(s) \, ds \leq 0\text{ a.e. }t \in (0,1)$$ $$x(t) \geq 0 \text{ a.e. ...
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How can I solve a nonlinear optimization problem where constraint contains exponential term?

I have the optimization problem as below: $\hspace{10mm} \text{Maximize} \sum_{k} \alpha_k {R}_k $ $ \hspace{10mm}\text{subjcet to:} $ $ \hspace{10mm}\exp \left[ - (2^{{R}_k } -1) \left( ...
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36 views

Soviet Optimization books

I am aware of an answer on Soviet math books here: Soviet Russian Mathematical Books and the book by Boris Polyak on non linear optimization. I am also aware of a few books by Kantorovich which I do ...
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17 views

Optimization problem: solving one implies solving the reverse?

I am looking to solve an optimazation problem $Maximize_{x} [A(x)]$ s.t. $B(x)\geq B_0$, where $B_0$ is a constant. If I solve this problem (i.e, finding the optimal $x^*$ that optmize while ...
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16 views

Classifying stationary points without the Hessian

Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$ be $\mathcal{C}^{\infty}$ in $\mathbb{R}^n$. I can calculate the gradient which results in an expression of the form $ \nabla_{\mathbf{a}} ...
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1answer
19 views

Maximum occurring at all points in a set

Is there any term for "sets where maximum of a mathematical expression in attained"? I just want to know if the set has any specific name. The set is infinite (do not consider discrete points). The ...
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29 views

Who knows Krotov's Method in Optimal Control Theory

I'm finishing my PhD thesis about applications of optimal control theory in the field of energy harvesting. In the course of my PhD I dealt with different ways to compute optimal controls, and I found ...
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1answer
36 views

Existence of minimum in bounded but open set

According to the Extreme Value Theorem, a continuous function achieves at least one minimum and one maximum whenever the set is bounded and closed (i.e. compact). In my case, I have a bounded and ...
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65 views

If $x,y,z>0$ and $xyz=32,$ Then the minimum of $x^2+4xy+4y^2+4z^2$ is

If $x,y,z$ are positive real no. and $xyz= 32\;,$ Then Minimum value of $$x^2+4xy+4y^2+4z^2$$ is $\bf{My\; Try::}$ Here I have Used $\bf{A.M\geq G.M}$ Inequality So $$\displaystyle ...
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69 views

Lagrange's multiplier not working

Given the function $f(x,y):=xy+x-y$. Let $D:=\{(x,y)\in\mathbb{R}^2:x^2+y^2\leq25\wedge x \geq 0\}$. Find the absolute maximum and minimum of $f$ on $D$. My working is as follows: $\begin{array} ...
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70 views

What is $H^\infty$ norm and why is it used in control theory?

Can anyone knowledgable elaborate on what exactly is a $H^\infty$ norm and why it is used in control theory instead of some other norms? Thanks!
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12 views

Finding maximum of a function represented by a back-propagation neural network

First, I train a standard feed-forward neural network over a training set of data points. I get an approximate function, say $F(x)$, represented implicitly by that neural network. Now I want to find ...
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38 views

Global maxima/minima of $f(x,y,z) = x+y+z$ in $A$

Find the global maxima/minima of $f(x,y,z) = x+y+z$ for points inside of $A = \{ (x,y,z) \in \mathbb{R}^3: x^2-y^2 = 1 \wedge 2x+z = 1 \}$ I renamed the conditions of $A$ to a function $g(x,y,z) ...
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65 views

Maximization of a nasty Gaussian likelihood

I have a Gaussian likelihood function, $$p(y|x) = \mathcal{N}(y; Ax, (x^\top V x + \lambda) \otimes I)$$ where $A,V,\lambda$ is known, and $\otimes$ is the Kronecker product. (the notation indicates ...
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14 views

$\left\{x^k\right\}$ converges to $x^*$ superlinearly iff $\left\|\nabla^2f(x^k)^{-1}\nabla f(x^k)+x^{k+1}-x^*\right\|=o(\left\|x^{k+1}-x^*\right\|)$

Let $(x^k)_{k\in\mathbb N}\subseteq\mathbb R^n$ be convergent to $x^*$. We say, that the convergence is superlinear iff $$\left\|x^{k+1}-x^*\right\|=o\left(\left\|x^k-x^*\right\|\right)\tag{1}\;.$$ ...
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“Optimaly” reordering the vertices of a hypergraph.

I am not even sure of how to search for an answer to this, or how to approach the problem myself, so I thought I would try to ask it here. Consider an n-vertex hypergraph where the vertices are ...
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19 views

Condition for product of increasing and decreasing functions to be quasiconcave?

Is there any condition for product of increasing and decreasing functions to be quasiconcave? More specifically, I am having in mind a condition for $F(x)\cdot(1-G(x))$ to be quasi concave where $F$ ...
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93 views

Trace minimization subject to diagonal constraints

Problem Revisited - Edited for conciseness: We are given two set of data points X [$p \times n$] and Y [$q \times n$]. Let us assume $X = \hat{X} + \tilde{X}$ and $Y = \hat{Y} + \tilde{Y}$ I am ...
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13 views

constrained optimization including sum of two upper incomplete gamma function in both fitness function and constraint

i'm trying to solve this constrained optimization problem the constraint is $$\zeta=\frac {\Gamma \left(M,\frac {\lambda}{\left |\sum_{i=1}^Nw_i*h_{ei} \right|^2 + \sum_{i=1}^N\left |w_i \right|^2} ...
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27 views

Classifying Critical Points of $f(x,y)=xy-x+2x^3-yx^3$

I am classifying the critical point(s) of $ f(x,y)=xy-x+2x^3-yx^3 $: I first found the critical points by solving for $ f_x=f_y=0 $: $f_x= y-1+6x^2-3yx^2=0 $ $f_y= x-x^3=0$ Hence $x=0$ and ...
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443 views

Absolute extrema

Find the absolute extrema of the function $f(x,y)=2xy-x-y$ over the region of the xy-plane bounded by the parabola $y=x^2$ and the line $y=4$ I was wondering if I needed to use Lagrange multipliers to ...
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1k views

Are triangles the strongest shape?

They are according to the buzz on the Internet (and most stable too), despite competition from circles. Mythbasters even proclaim that "triangles are the strongest shape because any added force is ...
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Approximate Dynamic Programing - Discount Factor for Very Long Horizons

I want an optimal strategy for a very long time horizon, say $K=100000$. I have dynamic decision making problem where next state $x_{k+1}$ is determined by the probability distribution ...
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61 views

Optimizing over intersection of polytopes inside polytope

I have the following situation, as shown in the following diagram: $W=\{w_i\}_{i=1..|W|}$ is a set of vertices (show in diagram in blue) located within a regular simplex and having coordinates $\in ...
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27 views

Minimise the result of an expression

What is the minimum value the expression ${a} + 3{b} + 3{c} + {d}$ can have if $$a, b, c, d \in \mathbb{N}$$ $${a} \neq {b} \neq {c} \neq {d}$$ and the sum of any two variables is not equal to ...
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44 views

Why would integrating acceleration give the following solution?

Suppose I have a mass with equation of motion described by: $x^{''}(t) = F(t) - 1$, $0<t<T$, all initial conditions equal to zero $F(t)$ is some unknown force My text claims that the equation ...
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Representing multivariate optimization problem as unconstrained single variable optimization

I have a function $f(x,y)$ that I must optimize (max and min) on G={$(x,y)|x+y=9$} I am asked to represent the problem as an unconstrained single variate optimization problem. I'm really not sure ...
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1answer
13 views

Conditions for a smooth optimizer?

Consider a function $f:\mathbb{R}^n\times\mathbb{R}^m\to\mathbb{R}$. I am trying to determine conditions (on $f$ and/or $X$) under which the maximizer defined by \begin{align} \hat x(\alpha) = ...
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Showing a $\mathbb{R}^3 \rightarrow \mathbb{R}$ function attains a global minimum at the origin without using calculus.

Given $f: \mathbb{R}^3 \rightarrow \mathbb{R}$ defined as $$f(x) = x^2 + 3y^2 +2z^2 - 2xy + 2xz$$ I am trying to show $f$ attains a global minimum at the origin without using calculus. I was thinking ...
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Need help figuring out routing problem

Thanks in advance for helping me with this routing problem. It's for a digital instrument I'm building, six sine-wave oscillators that feed back into each other in a kind of recursive web. Here's ...
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23 views

$F(x,y)=2x^4-3x^2y+y^2$. Show that $(0,0)$ is local minimum of the Reduction of F for every linear line that passes through $(0,0)$.

first of all I checked if (0,0) is critical point $Df(0,0)=(8x^3-6xy,-3x^3+2y)| = (0,0) $ now my idea was to replace $y$ with $xk$ because of the reduction of $F$ ,and find the hessian matrix to ...
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Positiveness of a real valued function of $n$ variables??

Let $\{a_1,...,a_n\}$ be a set of $n$ non-negative parameters, we define $x^*=(x_i^*)_i$ as the $n$ dimensional vector with components: $$x_i^*=\frac{a_i^2}{\sum_j a_j^2}$$ Let $F:\Delta\to ...
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7 views

Non linear Optimization for resource allocation

I want to maximize the sum rate of a wireless system while maintaining fair allocation by using fairness constraint. $R_k$ is the rate for each user. I have set up my objective function as : Maximize ...
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1answer
18 views

Sufficient conditions for the Hardy-Littlewood Maximal function $M(f)$ being continuous

There are four common versions of Hardy-Littlewood Maximal operator $M(f)$: centered/uncentered + ball/cube. I noticed that the continuity of $M(f)$ depends on the version. For example, let $f$ be the ...
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Get number equation using specific set of values for get given answer

I have do it for AI assignment. Need a logic for finding solution ..Here is the explanation of problem . I have answer ( any number like for example 10 ). And have some set of numbers (like for ...
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2answers
28 views

Showing a $\mathbb{R}^2 \rightarrow \mathbb{R}$ function attains a global maximum

Given $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ defined by $$f(x,y) = (ax^2+by^2)\exp(-x^2-y^2)$$ where $a > b > 0$, how can I show $f$ attains a global maximum? It is easy to show that it ...
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Definition issue with limiting directions

In Nocedal/Wright's Numerical Optimization (1999) in section 12.3 the notion of feasible sequences and related limiting directions are introduced as a starting point for the proof of the ...
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Gaussian likelihood - test two observations for same parent population

If I have an observation $x$ with a Gaussian distributed observational error of standard deviation $\sigma$ then the sum of likelihoods of that observation having the error free values $x_1^{\prime} ...
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Weight Distribution Optimization

I have a set of buckets all of equal capacity. I have a set of equal size balls of varying known weights. Each bucket must contain the same number of balls. How do I distribute the balls such that the ...
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34 views

Optimal solution in which only one decision variable is non-negative

Given the following LP: \begin{align} \max\quad & 29x_1 - 4x_2 + 5x_3 + 7x_4\\ \mathrm{s.t.}\quad & 4x_1 - x_2 + x_3 = 1\\ &3x_1 - x_2 + x_4 = 1 \end{align} show that an ...
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334 views

Baseball Roster Optimization

I'm trying to programmatically optimize a Fantasy Baseball Roster that requires a fixed number of players at position (2 Catchers, 5 Outfielders, etc.) and has a salary constraint (total draft price ...
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May be a trivial question regarding constrained optimization

Optimization problem is to find $x$>0 which $min \ \ L=\frac{A\left ( B(\frac{C}{Cx-B}+\frac{1}{x})+2C\log(\frac{B}{x}-C) \right )}{B^3}$ $s.t \ \ x\leq K $ Rewriting the objective ...
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Stochastic Control

I would like to solve the following stochastic dynamic programming in the discrete-case and continuous case: Let the state variables have the following dynamics: \begin{align*} dS_t = \mu S_t dt + ...
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54 views

Optimal algorithm for guessing random variable

Let's say you have some unknown quantity $$X\in [0,1]$$ We have N tries to guess the value of X - if you guess $$g_{i}\le X$$ then you capture value $$V_{i} = g_{i}$$ while if your guess is over the ...
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How are the tolerances evaluated in fmincon? specific/complete mathematical formulations needed.

I'm currently studying the stopping criteria about fmincon using different algorithms and I'm wondering how are the tolerances are actually evaluated and compared in the built-in function fmincon. ...
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1answer
26 views

Sequential information discovery in minimum number of steps when some items have information about other items

There are N items, say three: call them A B and C. For each item, there is an associated bit (0 or 1) and there is a prior probability that the bit is 1, call them p(A), p(B) and p(C). There is some ...
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708 views

Finding the center of mass of a cylinder

Help finding center of mass of soda can? If you represent the soda can as a right-circular cylinder radius=$4$ cm height =$12$ cm We are told to neglect the mass of the can itself. When the can ...
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41 views

what is the upper bound of $\max \mathbf{w}^T\mathbf{x}_i$

I need to find an equation for the upper bound of $\max \mathbf{w}^T\mathbf{x}_i, \; i=1, \dots N$. where $\mathbf{w}$ and $\mathbf{x}_i$ are two vectors. I need to find a function $f$ which holds ...
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33 views

Optimize $\max _{x_1,x_2,…,x_N} N , \text{ s.t.} \sum_{i=1}^N f(x_i) \le a$

$Is there general theory for solving optimization problem of the following kind \begin{align} &\max _{x_1,x_2,...,x_N} N \\ \text{ s.t.}& \sum_{i=1}^N f(x_i) \le a\\ &\sum_{i=1}^N g(x_i) ...