Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [optimization]

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.

In mathematics, computer science, economics, or management science, mathematical optimization (alternatively, optimization or mathematical programming) is the selection of a best element (with regard to some criteria) from some set of available alternatives.

An optimization problem can be represented in the following way: given a function $f:A\to\mathbb{R}$ from some set $A$ to the real numbers, we want to find an element $x_0\in A$ such that $f(x_0)\le f(x)$ for all $x \in A$ ("minimization") or such that $f(x_0)\ge f(x)$ for all $x \in A$ ("maximization").

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How to maximize a function when you cannot solve gradient = 0

I have to maximize this function : $$ f(x,y) = a\sqrt{x} + b\sqrt{y}$$ $$ a, b \in{R+*} $$ Knowing that $$ 0 ≤ x ≤ 2 − 2y $$ with $$ 0 ≤ y ≤ 1 $$ I said that f is a linear combination of 2 concave functions so it has a maximum (for 0<=x<=2-2y and…
Valentin
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Homework - Getting Started - Optimization

I am starting up my Masters after being out of school for awhile and this is my first week back. I am recalling how to complete a problem and I am wondering if anyone can help me get started. I do not need a final answer, just steps needed to…
Austin Gaylor
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the implication of $[Df(y) - Df(x)] (y-x) \le 0$

Let $\mathcal{D} \subset \mathbb{R}^n$ be a convex set. Let $f: \mathcal{D} \to \mathbb{R}$ be a differentiable function. Show that the following are equivalent: (a) $f$ is concave on $D$. (b) $f(y) -f(x) \le D f(x) (y-x)$ for all $x, y \in…
shk910
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Minimum Perimeter of Pentagon given area

A pentagon is made by mounting an isosceles triangle on top of a rectangle. What dimensions minimize perimeter $P$ for a given area $K$. This was asked in a test today. Let $x$ and $y$ denote the sides of the rectangle, and let $z$ be the height of…
Gagan Surathkal
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Is there any convex tight upper bound (major) for $f(\mathbf{x},Y) = \mathbf x^T Y \mathbf x$?

In a minimization problem, I have this expression in my objective function $$f(\mathbf{x},Y) = \mathbf x^T Y \mathbf x,$$ for $(\mathbf x,Y)\in \Omega = \{Y\succ 0,x\neq0\}$. So the function is biconvex, but not convex as a whole, (one eigenvalue is…
NSan
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How not to overfit validation set while tuning hypeparameters.

I am testing bayesian optimization for the purpose of hyperparameter tuning. For this purpose I divided my data into $3$ separate sets, one used for training, the second used for parameter tuning, third used for computing metrics. I noticed that for…
Slajni
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How to choose the $k$ points that given a cost matrix with other points, give the minimum sum of distances

For two sets of $n$ dimensional points this is their distance matrix: $p_{11}$ $p_{12}$ $p_{13}$ $p_{21}$ 1 2 3 $p_{22}$ 2 5 3 $p_{23}$ 2 3 1 $p_{24}$ 2 1 8 How can I get the $n$ points of $p_{1i}$ that give the minimum sum of…
Mario
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When is it equivalent to do a joint optimization over two variables in sequence over each variable?

Say I have a function $f(x,y)$. When is $$\min_{x,y}f(x,y)=\min_x\min_y f(x,y)=\min_y\min_x f(x,y)$$ and what happens when there are constraints?
Undertherainbow
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What to do if in an optimization problem with 2 variables and 1 equality constraint the determinant of bordered hessian is equal to zero?

I have to minimize $4x+y$ subject to $\sqrt{xy}=200$, and I got a critical point $(100,400)$ with $\lambda = -4$. It seems that it is the only critical point, but the determinant of bordered hessian at this point is zero and I’m confused now.
PupaLupa
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Minimizing quadratic forms and choosing sufficient step size

I'm trying to minimize the following: $P(x_{k+1}) = f(x_k) + \nabla f (x_k) \cdot (-h_k \nabla f(x_k)) + \frac{1}{2}(-h_k \nabla f(x_k))^T Hf(x_k) (-h_k \nabla f(x_k))$ I know that minimizing the quadratic form is by solving $Ax = b$ i.e. $x =…
LetmeKnow
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Backtracking line search: Are the bounds on $\alpha$ $(0,1)$ or $(0, 1/2)$?

In my optimization class we are using the Boyd and Vandenberghe Convex Optimization textbook, which gives the following stopping criterion for backtracking line search (p. 464): $$f(x + t \Delta x) \leq f(x) + \alpha t \nabla f(x)^T \Delta x$$ Here…
Max
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Is there a proof that simultaneous optimization delivers better results then sequential optimization for non-separable, multivariable functions?

$f(x,y)$ is a non-separable, non-negative real-valued function and $f(x,y)$ should be maximized over $x$ and $y$. Is there a proof that the simultaneous maximization $$\max_{x,\ y} f(x,y)$$ delivers better results then the sequential maximization?…
sebgrocp
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Quadratic optimization problem with an orthogonality constraint

$X \in \mathbb{R}^{n\times d}$, $P \in \mathbb{R}^{n\times k}$ and $L \in \mathbb{R}^{k\times d}$. What is the best way to solve a simple quadratic optimization problem with an orthogonality constraint: $min_{_{P}}(\left \| X - PL \right \|_{F}^{2})…
Michael
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Optimization problem (rectangle inscribed in a right triangle)

I am stuck on this optimization problem: I am supposed to find the largest rectangle that can be fit into the right triangle. I think I am having trouble with setting up the constraints. I am able to optimize cylinders in spheres but I cannot seem…
Kot
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Infeasible and Feasible solution of polyhedron

Let $P:= \{Ax = b, x \geq 0\} $ be a polyhedron in standard form, where $$A = \begin{bmatrix} 1 & -2 & 1 & 0 & 0\\ 2 & 1 & 0 & 1 & 0\\ 0 & 1 & 0 & 0 & 1\\ \end{bmatrix} \text{ and } b = \begin{bmatrix} 4 \\ 18 \\ 10 \end{bmatrix}$$ Find two…
holala
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