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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Local extreme value & saddle point: multi variable calculus

I am asked to find all local extreme values & saddle points of $$f(x,y) = 2x^2 + y^2 - xy - 7y + 8$$ $$f_x(x, y) = 4x-y, \qquad f_y(x,y) = 2y-x-7$$ $$f_x(x,y) = 0, \qquad y = 4x$$ $$f_y(x,y) ...
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Lp optimal solution question

i have a general question. if there is a general LP problem $c^Tx$ s.t $A\cdot x \le b$, and $x \ge 0$ and assuming that the components of $c$ are non-zero entries then how can I prove that when $x$ ...
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Minimum ladder over wall optimization

A fence 6 feet tall runs parallel to a tall building at a distance of 2 feet from the building. We want to find the the length of the shortest ladder that will reach from the ground over the fence to ...
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Split a set of numbers into 2 sets, where the sum of each set is as close to one another as possible

Given a set of numbers, I'd like to split this set into 2 sets, where the sum of each set is as close to equal as possible. How would I go about doing this in a programmatic way? Thanks in advance ...
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Optimization of relative entropy

Wondering if my following question is an application of information theory: Lets say we have a factory and ship boxes of stuff outside. If a competitor stands outside my factory, observes the stream ...
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396 views

Find extremes of function $f(x,y,z) = x^2y + y^2z + x - z$

I am preparing for an exam tuesday morning and I would like to ask you, if someone could please review my solution for the following excercise. I don't have the correct answer so I am unable to check ...
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585 views

Find minimum in a constrained three-variable equation

After my last question I have worked through the math quite a bit and now I'm stuck again. This time my question is less wordy. I have two equations for $t$, one with respect to each $a_{x}$ and ...
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Proving that $\text{ri rge}\,A=\text{ri conv rge}\,A$

"If $A:\mathbb R^n\rightrightarrows\mathbb R^n$ is maximal monotone,then $\text{ri rge}\,A$ is convex". This is a proposition in auslender's book about the asymptotic cones. We can prove that ...
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249 views

Minimize Frobenius norm with unitary constraint

I am trying to find a unitary tramsformation, $M$, that minimizes $\Vert MA-B \Vert_F^2$ where $A$ and $B$ are $N\times L,\;L\ge N$. I know how to solve it without the unitary constraint. I thought ...
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Optimizing non linear programs of two variables

The scenario is; We've got $n$ stationary 360$^{\circ}$ sensors in an confined area (each sensor is located at some arbitrary $\left(x,y\right) = \left(x_{n},y_{n}\right)$), once a unit $t$ enters ...
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Approximate a function over the interval $[0, 1]$ by a polynomial of degree $n$ (or less).

To approximate a function $G$ over the interval $[0,1]$ by a polynomial $P$ of degree $n$ (or less), we minimize the function $f:R^{n+1} \to R$ given by $F(a) = \int_0^1 (G(x) - P_a(x))^2\,dx$, where ...
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Prove $ax - x\log(x)$ is convex?

How do you prove a function like $ax - x\log(x)$ is convex? The definition doesn't seem to work easily due to the non-linearity of the log function. Any ideas?
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241 views

Computing the volume of a set efficiently

Given a set of vectors $\mathbf v_i$ for $i=1,\dots,k$, $\mathbf v_i \in \{0,1\}^n$, is that possible to efficiently find the volume of the set, $$\left\{\mathbf x \in [0,1]^n:\mathbf x \le ...
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Is $f(t)=t^\alpha$ for $\alpha\in(0,1)$ a sub-additive function? [duplicate]

Possible Duplicate: Does $|x|^p$ with $0<p<1$ satisfy the triangular inequality on $\mathbb{R}$? Is the function $$f(t)= t^{\alpha},\quad \alpha\in (0,1)$$ a subadditive ...
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Minimizing weighted sum of distances

given a set of coordinates and the following function: cost = $\sum \sqrt{(x_i−X)^2+(y_i−Y)^2}w_i$ I would like to find the point (X, Y) for which this function is minimal. A simple example shows ...
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636 views

Question about the simplex method complexity

So I know that in general the simplex method for linear and convex quadratic programming can require exponential time. But assuming a positive semidefinite quadratic program that is solvable by the ...
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687 views

Optimally combining samples to estimate averages

Suppose I have two tables, each of unknown size, and I'd like to estimate the average of their true sizes. I hire 2 contractors: one guarantees good precision (i.e., her measurement ...
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Online stochastic convex optimization.

I need to find/approximate the argument that minimizes a stochastic convex function $F(\theta, Z)$: $$ {\arg\min_{\theta}} E_{Z}[ F(\theta, Z) ]$$ Where $Z$ is some random variable (we could assume ...
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406 views

How to maximize area of two circles inside a rectangle without overlapping?

Two circles have to be drawn inside a rectangle of dimensions $W\times H$ such that the area of both circles is to be as large as possible without overlapping. Let the radii of the circles be $r_1$ ...
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A primitive function of $ e^{x^{2}} $

I made some efforts to set a closed form of primitive function of $ e^{x^{2}} $ i find this function : $ f(x)=\frac{x}{2x^{2}-1}e^{x^{2}} $ where : $f'(x)=(\frac{x}{2x^{2}-1}e^{x^{2}})'$= ...
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Minimizing Frobenius norm for two variables

I need to minimize squared Frobenius norm: $\|\mathbf{A} - \mathbf{x}\mathbf{y}^T\|_F^2$. Namely I need to prove that for this norm to reach minimum $\mathbf{x}$ should be eigenvector of ...
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Optimization with a constrained function

Okay so I understand how to find points of extrema when for example, We have $3x^2 + 2y^2 + 6z^2$ subject to the constaint $x+y+z=1$. I followed the method of the Lagrange multiplier and resulted in ...
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Nonconvex set converging to a convex set despite holes

I'm looking at the example in Figure 4-7 of "Variational Analysis" (Rockafellar and Wets). Basically, there's a sequence of sets $C_{\nu}$ riddled with holes, and it states that the sequence ...
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significance of zeros of a transfer function?

In control theory, the poles of a transfer function give information about the stability and behavior of a system. I'm not sure and can't find anywhere what the significance of the zeros of a ...
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Find the maximum or minimum value of the quadratic function.

Find the maximum or minimum value of the quadratic function by completing the squares. Also, state the value of $x$ at which the function is maximum or minimum. $y=2x^2-4x+7$ $x^2$ has a coefficient ...
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How to maximize (baking) surface area?

I like eating crust, so I am trying different baking molds to try to get the most crust per dough. More generally, I'm interested in the reverse of this more specific question — how to maximize the ...
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682 views

Compute the minimum distance between the centre to the curve $xy=4$.

I wish to solve the following problem: Compute the minimum distance between the center to the curve $xy=4$. But I don't know where to start from?
3
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112 views

Effecient way to find optimal solution in a 2 player game

I have a function: \begin{equation*} f(a_1,\ldots,a_7,b_1,\ldots,b_4)=-14-7 a_1+30 a_1 a_2-7 a_4-2 a_4 a_5+21 a_6+21 a_7+16 a_1 b_1-24 a_1 a_2 b_1+6 a_4 b_1-6 a_4 a_5 b_1+6 a_1 b_2-6 a_1 a_2 b_2+8 a_4 ...
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How many points to find a polynomial?

I would like to fit a formula $ax^b + cx^d+ e$ to a set of points. I have two questions. If my data were perfect, how many points do I need in the worst case to get $a,b,c,d,e$ exactly? If my data ...
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Shortest distance between two shapes

This is the scenario of my problem. I have an image of two objects ( of arbitrary shape, not convex, not touching or crossing each other, kept a few space apart). And I am supposed to find the ...
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Maximum of an expression with four variables

Assume $0 < p_1 \le p_2\le p_3 \le p_4$. What is the maximum of the following expression? $$ \frac{\left(p_1+p_4\right)\left(p_2+p_3\right)}{\left(p_1+p_3\right)\left(p_2+p_4\right)} $$ Is that ...
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What is the complexity of computing the minimum distance between two convex polyhedra that both have $n$ faces?

EDIT: (in response to what deinst said) sometimes using a sledgehammer for some menial task is rather convenient - especially if it also has the complexity $O(n)$ (which is what my question is about) ...
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Least Square Method with Positive Parameters

this is my first post here in the Stack Exchange. A friend told me about this forum and I'm giving it a try. I searched a bit past threads, but couldn't find what I wanted, so I'm posting the problem ...
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Derivative of function including matrix logarithm

Is the following equation a first order approximation or incorrect for general matrix Lie groups? And what are the higher order terms? $$\frac{\partial}{\partial\mathbf x} (\log(\mathtt ...
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Lagrange multipliers with non-smooth constraints

I read in a textbook a passing comment that Lagrange multipliers are not applicable if there are points of non-differentiability in the constraints (even if the constraints are continuous). For ...
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linear least squares minimizing distance from points to rays - is it possible?

I'm writing a tool whose purpose is to process data from a sensor that provides the true bearing to a target, and combine measurements taken at various times into an estimate of the target's position ...
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Formulating an LP problem with vectors

We have $m$ vectors $v_1,v_2,\dots,v_m\in\mathbb{R}^n$ and $m$ numbers $t_1,t_2,\dots,t_m\in\mathbb{R}$ and we want to find a vector $y\in\mathbb{R}^n$ such that $$|v_i^Ty-t_i|\leq D$$ for ...
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What's wrong in this dual derivation?

I have a function in the form \begin{align} f(q,M)=\sup_{0\leq \alpha \leq 1} -\alpha^T (R\odot M)\alpha+\alpha^Tq \end{align} which is a dual of a minimization problem, where $R$ and $M$ are ...
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Optimization by Symmetry?

Let $$f(x,y,a,b) := \frac{xa+yb}{\sqrt{xa^2+yb^2}},$$ where $x,y,a,b$ are all positive. Define $$g(a,b) = \min_{x+y=1,\,x,y\ge 0}f(x,y,a,b).$$ How would one solve for $g(a,b)$? I have solved this by ...
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binary quadratic optimization problem

I am trying to solve the following binary quadratic program. $$ \min_{\Delta} \Delta^T H \Delta + c^T\Delta \\ \text{Such that:} ~~~\Delta\in \{0,1\}^n ~~\text{and}~~ \sum_{i=1}^n \Delta_i \leq \Gamma ...
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Is the opposite of the Second Derivative Test also true?

Given the Second Derivative Test, one case says : If $f(x_0)''<0$, then $f$ has a local maximum at $x_0$. Is it also true that, if $f$ has a local maximum at $x_0$, $f(x_0)'' < 0$ ?
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What would be the objective functions for this problem?

I have the following data (this is just a sample of my entire dataset): # Distance PriceIndex Rating 1 400 3 5 2 420 2 4 3 500 1 2 Considering the ...
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Lagrange multiplier expression

I would like to solve the following optimization problem using the gradient ascend method: \begin{array}{ll} \text{maximize}_{\theta} & \theta^TQ_1\theta + b_1\\ \text{subject to} & ...
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How to maximize an entropy function?

I'm very novice in optimization and have a convex optimization function of form $\sum_{i,k} p_{k,i}*\log{p_{k,i}} $ to minimize with the following constraints: $\forall i, a_i = \sum_{k=1}^{m} b_k. ...
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Max perimeter of triangle inscribed in a circle

What is the maximum perimeter of a triangle inscibed in a circle of radius $1$? I can't seem to find a proper equation to calculate the derivative.
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How to maximize $\left({a+b \choose a} 2^{-a-b}\right)$?

How can you maximize $\left({a+b \choose a} 2^{-a-b}\right)$ assuming, $a,b \geq 0$ and $0< (a+b) \leq n$, where all the variables are non-negative integers? Is the maximum when $a=b=n/2$, ...
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Previous table of simplex algorithm

I have this problem of simplex method which shows cycle . Problem : Maximize $Z=10x_1-57x_2-9x_3-24x_4$ Constraint to : $ 0.5x_1-5.5x_2-2.5x_3+9x_4<=0 $, $ ...
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Minimisation of a distance sum

I have a list $L$ of $N$ numbers, and I want to choose $k$ numbers $\{x_1,x_2, \ldots,x_k\} \subseteq L$ in such a way value $S$ of the those K numbers is minimum. $$ S = \sum_{0< i < j <= k} ...
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Minimize $\|Ax-b\|$ where $x$ is a binary vector

For a software project I'm involved on, I have a situation where I have a large vector that is the sum of some smaller vectors. I know all the possible small vectors, and I know that no two of them ...
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Simultaneous Maximization and Minimization

I have a function with two variables say $$g(x,y)=f(x)−h(x,y)\ $$ where $$f(x)= ax-bx^2\ $$ and $$h(x,y)=(x+y)^2\ $$ and $$ y\ge0, x+y\ge0.$$ My purpose is to maximize $g(x,y)$ for $x$, ...