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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Optimizing an expression containing sum of square roots of squared terms

For optimization problems involving square root, it is common to optimize the squared expression instead of that containing the square root. What if we have sum of squared expressions ? Consider the ...
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71 views

Convex optimization approximation

Consider the optimization problem $\mathcal{P}_0$ $$ \min_{x \in \mathbb{R}^2} \left\| x-p \right\|^2 $$ $$ \text{sub. to: } \ A x \leq b, \ \ x_1^2 + x_2^2 = 1 $$ where $p \in \mathbb{R}^2$ is a ...
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50 views

minimizing sum of different least squares?

Can we write the minimization problem: $$\operatorname{min}\limits_{x\in\mathbb{R}^n}\sum_{i=1}^{n}\|C_i x-b_i\|_2^2$$ as a least square problem?
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44 views

Global/local optima for this function

I have the following function $f(x_1,x_2) = \frac{x_1}{x_2+p} + \frac{x_2}{x_1+p}$ where $x_1$ and $x_2$ $\in$ $[0,1]$ and $p > 0$ is a constant I want to find global/local maxima for this. ...
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165 views

How exactly do I prove that I find the maximum of the function

I am currently trying to maximize an objective function $f(a,b,c,d,e)$ over the variable $b$ only. By taking the derviative of f over b, setting it to zero, I can solve b in terms of the other 4 ...
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30 views

Minimal volume of a tetrahedral

I'm unsure how to solve the following problem: Let $\textbf{p}=(a,b,c) \in \mathbb{R}^{3}$ with $a,b,c > 0$. For $\alpha , \beta > 0$ the equation $$\alpha (x-a)+ \beta(y-b) + (z-c) =0$$ ...
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25 views

Minima point is a solution point

Consider $$f:\left[0, \dfrac{\pi}2\right] \to \mathbb R$$ defined as $$f(x)=\sup\{x^2,\cos x\}.$$ It is easy to show that $f$ has an absolute minimum point at $x_o \in I$ , but how to show that $\cos ...
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68 views

Shortest path problem: dual formulation and proof of total unimodularity

The IP formulation of the shortest path problem looks as follows: \begin{align*} \min & \sum_{u,v \in A} c_{uv} x_{uv}\\ \text{s.t } & \sum_{v \in V^{+}(s)} x_{sv} - \sum_{v \in ...
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49 views

Logistic function approximation of the real valued Riemann $\zeta(x)$ function

Given the function: $$f(x)=\dfrac{a}{1-b\exp(-cx)}+d$$ where: $a = 0.7071$, $b = 2.21$, $c = 0.7672$, $d = 0.2942$, I found the following inequality: $$|\zeta(x) - f(x)|\lt \epsilon$$ for ...
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Nonlinear Optimization problem

Function $f(x) \in \mathbb{R}^n$, $(n\geq 1)$, depend on one parameter $x \in \mathbb{R}$. Performing a nonlinear transformation of $f(x)$, we obtain function $g(y) \in \mathbb{R}^n$. This ...
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35 views

How to find all stationary points of $ \alpha\|v\|^2-\|x^Tv\|^2+\|g^Tv\|^2$

Let $v,x,g$ be three vectors and $\alpha$ be a constant. The problem is $$\min\limits_v \{\alpha\|v\|^2-\|x^Tv\|^2+\|g^Tv\|^2\}$$ where $\|v\|^2=\sum\limits_{i=1}^{|v|}v_i^2$ and $|v|$ is the ...
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67 views

Optimizing elementary symmetric polynomial on the unit sphere

I'd like to optimize $x_1 x_2 x_3 + x_1 x_2 x_4 + x_1 x_3 x_4 + x_2 x_3 x_4$ on the unit 4-sphere. I'm thinking I should do lagrangian optimization, but I'm having trouble solving the resulting ...
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77 views

optimization problem: finding an hyperplane separating one point from a set of pointy maximizing the distance

I have this problem: I have a set of n-dimensional points $P$. I have one more n-dimensional point $q$. The points in $P$ are linearly separable from $q$ (i.e. it always exists an hyperplane $n^t x ...
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45 views

Sufficient condition on convex function such that $f(x) > -\infty$ for all $x$.

Let $f : \mathbb R^n \to [-\infty, \infty]$ convex and let $f(\overline x) > -\infty$ for $\overline x \in \mbox{int}(\mbox{dom}(f)$. Show that $f(x) > -\infty$ for all $x \in \mathbb R$. ...
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41 views

The convexity of convex function's range

Given a convex function $f\colon X \to \mathbb R$ with convex domain $X \subseteq \mathbb R^n$, is the range of $f$ a convex set also?
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33 views

Hessian of non-differentiable function

Given a function $f = \max\{f_1,f_2\}$ with $f_1,f_2$ convex and differentiable, I know I can calculate the subgradient of $f$. Is there also an equivalent of the subgradient for the (sub)Hessian? ...
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33 views

Dual norm equivalence?

$\|\|$ is a norm in $R^n$, its dual norm is defined as $\|s\|^*=max_{\|x\|=1}s^Tx$. We denote $s^\#$ as any vector in the following set: [Arg $max_x: \ \ s^Tx-\frac{1}{2}\|x\|^2$] How to verify ...
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26 views

Is convex optimization belong to linear programing or nonlinear programing?

Is convex optimization belong to linear programing or nonlinear programing? Is convex optimization undergraduate topics or graduate topics?
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What exactly is non-convex optimization

I am coming across the term: non-convex optimization problem. What exactly is this non-convex structure, and how do I know by only looking at the structure of the problem, I could tell it is ...
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43 views

Why does the interpolation error go to zero if we increase the number of sampling points?

This question is motivated by polynomial interpolation. We know that for $f\in C^{n+1}[a,b]$ and $a=x_0<\dots<x_n=b$ holds $$\| f - p_n \|_\infty \leq \frac{1}{(n+1)!} \| f^{(n+1)} \|_\infty ...
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97 views

the coordinate axes has minimum area

How can i find the point in the first quadrant on the parabola $$ y = 4-x ^ 2 $$ such that the triangle tangent to the parabola at the point and the coordinate axes has minimum area. Some help to ...
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Regularization vs. Inequality Constraint

For what values of a regularization parameter $\alpha$, there is an equivalent inequality constraint in convex optimization? In particular, in the convex optimization problems below $$ \text{ Problem ...
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To prove the existing and uniqueness of a solution

Let function $f$ be differentiable and convex in $R^{n}$. How can it be proved that $\forall \lambda > 0$ solution of system equations $f'(x) = -\lambda x$ exists exclusively ($\exists \hspace{3mm} ...
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67 views

Given a polynomial of degree 5, get minimum and maximum without using derivatives

Given a quintic polynomial (in my case, $x^5+2x^4+16x-32$), I am supposed to get its maximum and minimum value for the interval $I=[-2;2]$ without using the derivative of the corresponing polynomial ...
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31 views

Substitute for constraints in constrained optimisation problem

I have to solve the following optimisation problem $$\max_{c,\ell, L} \{\ln(c) + \ln(\ell)\}$$ subject to the constraints: $c=Lw$ and $1=\ell +L$. Is it okay to solve the problem ...
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81 views

How to find center and radius of hand-drawn circle? [duplicate]

You are given a set of points {(X1,Y1), (X2,Y2),...} which represent a hand-drawn circle, so it's not perfect. You are asked to find the center and radius of this circle. My intuition tells me this ...
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224 views

How do you minimize “hinge-loss”?

A lot of material on the web regarding Loss functions talk about "minimizing the Hinge Loss". However, nobody actually explains it, or at least gives some example. The best material I found is here ...
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Distance from Ellipsoid to Plane - Lagrange Multiplier

Find the distance from the ellipsoid $x^2 + y^2 + 4z^2 = 4$ to the plane $x + y + z = 6$. I'm trying to do it using Lagrange multipliers over the distance equation, but then it just gets ...
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can I get help in solving this equation using simplex method big-M method

Objective: $\max Z= 100x_1+300x_2+400x_3$ s.t. $10x_1+20x_2+30x_3≤1600$ $\;\,\quad10x_1+15x_2+20x_3≤1500$ $\;\,\quad x_2+x_3≤50$ $\;\,\quad x_1+x_2+x_3=70$ $\;\,\quad x_1,x_2,x_3≥0$
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39 views

Is this a discrete time Lyapunov function?

I have an algorithm to optimize a process. It is a discrete time algorithm. Every iteration of this algorithm changes the state of the process. I found a function, say $f(s)$, where $s$ is the state ...
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40 views

Finding/approximating 2 unknowns using one equation

I’m doing experimental data in a chemistry lab and I have faced this mathematical problem at a point of my work. Hope you guys can help me with that. What would be the best way to find two constants m ...
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maximizing a quadratic objective subject to sum constraint

Is it possible to solve the following problem $$ \max \,{x^ \top }x \\[0.1in] st: \sum\limits_{i = 1}^n {{x_i} = 1} \\[0.1in] 0 \le x_i \le 1 $$ Intuitively it happens when any one of the n x's is 1 ...
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Find extrema on the interval

Problem Find the extrema of the function $$f(x) = cos^2(x)$$ on the interval $ [-4,4]$ I can differentiate and get $$f'(x) = -2 \sin(x) \cos(x)$$ And set that to zero, but I'm pretty sure that's ...
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90 views

Optimization problem (total distance from point on sphere to other points)?

Let $M_i(x_i, y_i, z_i)$ be a set of $n$ fixed points. Given their coordinates, find a point $M(x, y, z)$ which is on the sphere $x^2 + y^2 + z^2 = 1$ and has the minimal sum of distances between ...
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Finding local max and min of a transcendental function

So.. there's no way in heck i'll be solving the derivative of this algebraically -- what can I do? here's a hint from my professor.. Since the picture may be unclear, the function is $$f(x) = ...
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71 views

What trick to calculate this Frechet derivative?

Let $u(t) \in L^{2}(0, 1)$. I need to calculate the first and second Frechet derivatives of $$J(u) = \int_0^1 \left(\int_0^{t^3}u(s)ds\right)^2dt$$ I am completely at a loss here: I know several ...
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Minimizing the following objective function with matrices

Suppose $A$ and $B$ are known matrices, and we are to find matrix $X$ that minimizes the following function, $$\frac{1}{2}||X||^2+\frac{1}{2}||X^TAX-B||^2$$ Taking the relevant derivative w.r.t $X$ ...
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26 views

Finding minimum norm

Let $A$ be $k\times k$ positive symmetric matrix, $K$ is $k\times d$ full rank matrix with $d<k$, and $v\in\mathbb{R}^k$. I'd like to find $x\in \mathbb{R}^d$ such that $(Kx-v)^TA(Kx-v)$ minimum. ...
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186 views

What minimizes the Chebyshev Distance?

For an arbitrary number of dimensions, I know that the mean minimizes the distance using the L2 norm and that the geometric median minimizes the distance function using the L1 norm (though I have yet ...
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how to impose binarity constraint in a vector

This is part of a homework problem. In an optimization problem, I need to have a K dimensional vector S, such that each entry of the vector is either 0 or 1, and $l_1$ norm of S is <= K. I can't ...
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Lagrangian with inequalities

I have a toy question on SVM , where i have to find the weight $w$ by solving the Lagrangian multiplier method by hand . I know Lagrangain with equalities only . Here I have to deal with inequalities ...
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104 views

Verify by Second Derivative Test

$$A(x)=2\sqrt{x^2-16}+\frac14\sqrt{68x^2-x^4-256}\;,\;\; (4 < x < 8)$$ of which the derivative is: $$a'(x)=\frac{2x}{\sqrt{x^2-16}}+\frac{136x-4x^3}{8\sqrt{68x^2-x^4-256}}$$ I first had to ...
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131 views

Solution to a Quadratic Minimization with Norm Constraint

How do I solve the optimization problem \begin{align} &\min_{\mathbf{x}\in\mathbb{C}^N}\mathbf{x}^H\mathbf{A}\mathbf{x}+2\Re\{\mathbf{b}^H\mathbf{x}\} \\ \mbox{subject to }\\ ...
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64 views

Inequality involving max is confusing me.

I am trying to understand one line in a derivation here. Simply put, the statement is that: $$ max\{\theta \ f_1(x) + (1-\theta) \ f_1(y) \ , \ \theta \ f_2(x) + (1-\theta) \ f_2(y)\} \leq \theta \ ...
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86 views

Is the geometric-to-arithmetic function convex or concave?

Consider a vector $\mathbf{x} \in \mathbb{R}_{++}^N$. Also consider two functions, $g(\mathbf{x}): \mathbb{R}^N \rightarrow \mathbb{R}$, and $a(\mathbf{x}): \mathbb{R}^N \rightarrow \mathbb{R}$, ...
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186 views

Concave function divided by a convex function. What is the result?

Let us say that I have a function $f(x)$ that we know is a concave. And let us also say that we have another function $g(x)$ that is a convex. If I make a new function, $h(x) = \frac{f(x)}{g(x)}$, ...
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36 views

Finding local minimum under constraint

How to find the minimum of $f(x) = ||x-\mu||^2$, where $\mu = (1, 1)$ and $< x, \mu > = 0$ (the inner product is $0$)?
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34 views

Can we find an $n$ that minimizes this function?

If we suppose that we have positive integers $k$, $c$, and $v$, can we find the $n$ that minimizes: $$k^n \frac{\log{2^v}}{\log{v}}v^{\log_2{(k \cdot v \cdot c/n)}}$$
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87 views

Symmetric Positive Definite and Gradient Proof

I have the function $f(x)=\frac {1}{2} \mathbf x^T Q \mathbf x - \mathbf b^T \mathbf x$ where $Q$ is symmetric. I'm trying to show that solving $\nabla f(\mathbf x) = 0$ is equivalent to solving $Q ...
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67 views

The least value of $a + b + c$ where $a,b,c$ are not perfect squares but their pairwise products are

The positive integers $a$, $b$ and $c$ are all different. None of them is a square but all the products $ab$, $ac$ and $bc$ are squares. What is the least value that $a + b + c$ can take? A $14$ B ...