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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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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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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26 views

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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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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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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1answer
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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1answer
9 views

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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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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25 views

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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41 views

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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22 views

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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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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40 views

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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26 views

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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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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35 views

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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18 views

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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41 views

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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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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1answer
40 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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90 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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32 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) ...
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41 views

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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38 views

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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1answer
16 views

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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27 views

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
26 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
44 views

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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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 ...
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52 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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42 views

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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19 views

Quadratic Programming “big M” method

How does the optimization problem $$\min_{x,\eta} \frac{1}{2} x^TGx+x^Tc+M\eta$$ $$ s.t. Ax+\eta-b \geq0$$ $$\eta\geq0$$ look in standard form? What would the KKT look like? The problem is, that ...
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31 views

Estimating coefficients in a physical system based on observations

I have a physical system which can be modelled as $$Ax+By+C=0$$ I have thousands of measurements of $x$ and $y$ from the physical system (includes some noise). I want to optimize for $A$, $B$, and ...
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Max Min Problem based on inequalities

I am referring to the problem here I know that statement 2 is sufficient. Because when $ n = 0, p\leq 2.5 $ and when$ p = 0, n\leq 2.2222$ However, how does analyzing just these 2 scenarios (i.e. ...
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Optimization formulation for a dynamic system. Constructing constraints for a problem.

I am trying to formulate a problem that goes the following Min $f(.)$ This is a generalized objective function. Subject to, $x_{i}^{(t+1)} = x_{i}^{(t)} + r_{i}^{(t)} - x_{i}^{(t)}z_{i}^{(t)}$ ...
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170 views

Minimum of an apparently harmless function of two variables

I would like to prove that the minimum of the function $$ f(x,y):=\frac{(1-\cos(\pi x))(1-\cos (\pi y))\sqrt{x^2+y^2}}{x^2 y^2 \sqrt{(1-\cos(\pi x))(2+\cos(\pi y))+(2+\cos(\pi x))(1-\cos(\pi y))}} $$ ...
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17 views

Minimizing a quadratic form with orthogonality constraints

Suppose $A$ is an $n$-by-$n$ symmetric matrix, and I want to find $x_{1}$ and $x_{2}$ that maximize $x_{1}^{T} A x_{1} + x_{2}^{T} A x_{2}$ subject to the constraint that $x_{i}^{T} x_{j} = ...
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30 views

Minimizing sequences and topology (direct method)

To show the importance of the choice of the topology for the direct method we have been assigned the following exercise which I've not been able to solve due to my lack of understanding on how strong ...
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17 views

Quadratic optimization problem with quadratic equality constraint

I am trying to solve the following optimization problem: $$ \min_{x \in \mathbf{R}^2} \, x^T A x + b^Tx \quad \text{subject to $x^T J x = 1$} $$ where $A$ is a positive semi-definite $2 \times 2$ ...
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30 views

How do you go about solving partial differential equations for finding critical points in general optimization problems?

I was reading about partial second derivative test for optimization problems and I came across the example here. I saw the equations have yielded four critical points, but I wasn't able to find those ...
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23 views

The minimum of $\sum_{i=1}^m(c_ix-b_i)^2$

We know the minimum of $(c_ix-b_i)^2$ is $$x_i^*=\frac{b_i}{c_i}$$ How to show that the minimum of $\sum_{i=1}^m(c_ix-b_i)^2$ with $c_i \neq0$ is ...
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32 views

Need help using matlab optimization tools [closed]

I'm working on some project involving large scale matrix and i need your help to solve an optimization problem with matlab, the problem is the following: $ \min_L \{\alpha Tr(Y^{t}LY) + ...
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1answer
49 views

$\frac{1}{x+1}+\frac{1}{y+1} +\frac{1}{z+1}$ minimum value if $xyz$ =k.$x,y,z$ are positive reals.

$\frac{1}{x+1}+\frac{1}{y+1} +\frac{1}{z+1}$ minimum value if $xyz$ =k.$x,y,z$ are positive reals.I think the minimum should be when $x=y=z=k^{1/3}$. How do I show it? I tried to use AM-GM inequality ...
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12 views

Surface fitting: Where to start from?

Often, we deal with identification problems such as identifying the parameters $\alpha_i$ where $z(x) = f_{\alpha_i}(x,y)$, which means simply $z$ is a function of $(x,y)$ and the parameters ...