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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Integer (Binary?) Optimisation of a problem

got a question regarding maximal optimisation of a problem. Refer to the table below: $$\begin{array}{c|c|c|} & \text{Area A} & \text{Area B} & \text{Area C} & \text{Area D} \\ ...
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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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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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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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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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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
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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Minimizing the sum of the $4^\text{th}$ power of a matrix entries.

Consider a real $n\times n$ matrix $X$. Suppose I would like to minimize the sum of the squares of its entries as a penalty term in some convex minimization. I can write the term using the Frobenius ...
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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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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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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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66 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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$\left\{x^k\right\}$ 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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Placing Circles Onto Lines For Optimality

Suppose you have a yet to be determined number of vertical lines with length 50 on which you'd like to place as many circles as you can. Each circle is 10 units in diameter and its outside edge must ...
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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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1answer
64 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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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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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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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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Reference Request: Soft handed text on duality theory?

Can anyone recommend a text on duality theory which includes basic formulation of the primal and dual formulation and some introduction to minimax problems? Preferably having some computation in ...
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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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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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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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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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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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Particular map from a square to a parallelogram

I would like to present you a problem I have to solve. I don't think its solution is elementary, so any hint you can give me is really welcomed. Let's consider $Q_1$ the square in $\mathbb{R}^2$ of ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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) ...
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Proving inequality using Lagrange multipliers, somehow?

While going over assignments preparing for an upcoming exam, I noticed the question Prove that $x^{4} + y^{4} - 4b^{2}xy \geq -2b^{4} \text{ }\forall\text{ } x,y \in \mathbb{R}$ I had used ...
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Given model determine Least Squares estimate - is my graph correct?

Given the model $\alpha_1 I_k - \alpha_2 I_k^2 = I_{k+1}$ corresponding to measurements shown in the table below $k$ | 1 | 2 | 3 | 4 $I_k$| 0 | 1 | 10 | 80 | Determine the least squares ...
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Minimization of a multivariate quadratic equation

I am interested in the minimum of a general multivariate quadratic equation for non-negative real numbers: $$ \begin{aligned} & \underset{x_i}{\text{minimize}} & & ...
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What is the Dual of this particular Linear Program ( I get a weird Dual)

maximize $x_1-2x_2+3x_3-4x_4$ s.t. $x_1+x_2+x_3+x_4 = 20$ $x_1,x_2,x_3,x_4\geq 0$ The Dual can be found by transposing the constraint matrix and interchanging the objective function with 20 in ...
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Implementation of EM algorithm for Gaussian Mixture Models using Matlab

Using the EM algorithm, I want to train a Gaussian Mixture model using four components on a given dataset. The set is three dimensional and contains 300 samples. The problem is that after about 6 ...
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1answer
28 views

L1 minimization linear programming

So suppose we want to minimize the sum of the absolute errors $\sum\limits_{i=1}^m |b_i - \sum\limits_{j=1}^n a_{ij}x_j|$ with respect to $x_k$ where $k=1,...,n$ So to formulate this as a linear ...
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1answer
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What's the best way to optimize this energy function, and is it convex?

I have an energy function $E({\bf y})=||\,g({\bf Ay+c})-{\bf d}\,||^2_2 + ||\,{\bf y-e}\,||^2_2 + \alpha\,|{\bf y}|_1$ I need to minimize this with respect to $\bf y$, all other variables being ...
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1answer
25 views

Compute eigenvalues of Hessian = $\begin{bmatrix}a&1\\1&2\end{bmatrix}$ such that function is convex/eigenvalues $\geq 0$

The Hessian matrix is given to be $\begin{bmatrix}a&1\\1&2 \end{bmatrix}$ where $a$ is a real number. EDIT: So to find the eigenvalues I find the determinant which is ad-bc so 2a-1. So in ...