What is Constrained optimization: Definition and 19 Discussions
In mathematical optimization, constrained optimization (in some contexts called constraint optimization) is the process of optimizing an objective function with respect to some variables in the presence of constraints on those variables. The objective function is either a cost function or energy function, which is to be minimized, or a reward function or utility function, which is to be maximized. Constraints can be either hard constraints, which set conditions for the variables that are required to be satisfied, or soft constraints, which have some variable values that are penalized in the objective function if, and based on the extent that, the conditions on the variables are not satisfied.
I'm reading the book Deep Learning by Ian Goodfellow, Yoshua Bengio, and Aaron Courville, and currently reading this chapter on numerical methods--specifically, the section on constrained optimization.
The book states the following.
Suppose we wish to minimize a function...
Hello,
I am using the Lagrange multipliers method to find the extremums of ##f(x,y)## subjected to the constraint ##g(x,y)##, an ellipse.
So far, I have successfully identified several triplets ##(x^∗,y^∗,λ^∗)## such that each triplet is a stationary point for the Lagrangian: ##\nabla...
Summary:: Hi, this is an exercise from an algorithm course. I have been trying for hours but I have no successful ideas on how to solve it. I can only understand that DP is the correct approach, since Greedy method does not work.
Suppose you have *n* friends that wants to give you an amount of...
Note this is in our Lagrangian Mechanics section of Classical Mechanics, so I assume he wants us to use Calculus of Variations to solve it.
The surface area is fixed, so that'll be the constraint. Maximizing volume, we need a functional to represent Volume. This was tricky, but my best guess for...
Hi all,
I was working on a problem using Euler-Lagrange equations, and I started wondering about the total and partial derivatives. After some fiddling around in equations, I feel like I have confused myself a bit.
I'm not a mathematician by training, so there must exist some terminology which...
I have seen the implementation of L-BFGS-B by authors in Fortran and ports in several languages. I am trying to implement the algorithm on my own.
I am having difficulty grasping a few steps. Is there a worked out example using L-BFGS or L-BFGS-B ? Something similar to...
Problem:
Fix some vector ##\vec{a} \in R^n \setminus \vec{0}## and define ##f( \vec{x} ) = \vec{a} \cdot \vec{x}##. Give an expression for the maximum of ##f(\vec{x})## subject to ##||\vec{x}||_2 = 1##.
My work:
Seems like a lagrange multiplier problem.
I have ##\mathcal{L}(\vec{x},\lambda)...
Homework Statement
There is a typo in the problem, ”R > Σ n i=1 σi − n max 1≤i≤n σi” which should be R > n max (1≤i≤n) σi − (Σ n i=1 σi )
Homework EquationsThe Attempt at a Solution
Not sure where to go with part B or where to start...
Perhaps the title says it all, but I should expand it more, I guess.
So I am trying to explore more about constrained optimization. I noticed that there are very little to no formal (with examples) discussions on algorithms on nonlinear constrained optimization in the internet. They would...
Hi all,
I am looking for an efficient solution to solve the following problem. Can anybody help?
Assume a set S of elements ki and a set V of possible groupings Gj. A grouping Gj is a subset of S. Associate a weight wij to each mapping ki to Gj. The weights are infinite if ki ⊄ Gj, and finite...
Homework Statement
Homework Equations
Constrined optimzation
The Attempt at a Solution
("o" means dot product)
Let M={x|Ax=c} and f(x)=(1/2)x o Qx - b o x
Suppose x0 is a local min point.
Suppose, on the contrary, that x0 is NOT a global min point. Then there must exist a...
Homework Statement
Hello! I'm having some difficulty getting the objective function out of this question, any help/hints would be appreciated >.<
Company A prepares to launch a new brand of tablet computers. Their strategy is to release the first batch with the initial price of p_1 dollars...
Hi all,
I have this kind of optimization problem:
Variable to control: A=A=[a1;a2;...;am]
objective function to minimize: L=A*TL
where
L is a scalar
T is a matrix [1,m]
TL is a matrix [m,1]
constrain:
Dt>Dtv
where:
Dt=[dt1;dt2;...;dtn]
Dtv=[dtv1;dtv2;...;dtvn] is a...
Hi all,
I am working on a project and stuck at the following problem.
Find vector x_{n\times 1} which minimizes the function
f(x) = \sum_{i}^{n}x_{i}^{2}
subject to the linear equality constraint
[A]_{m\times n} x_{n \times 1}=b_{m\times 1} with m\leq n
The function f(x) trivially...
Hi,
I'm trying to do a constrained optimization problem. I shall omit the details as I don't think they're important to my issue. Let f:\mathbb R^n \to \mathbb R and c:\mathbb R^n \to \mathbb R^+\cup\{0\} be differentiable functions, where \mathbb R^+ = \left\{ x \in \mathbb R : x> 0...
hi
i want to find values of a,b,c such that..
Minimize (a+b+c)
constrained to
(x-a)^2 + (y-b)^2 + (z-c)^2 less than equal to R(z)
(x-a)^2 + (y-b)^2 + (z-c)^2 greater than equal to r(z)
can anyone help me solving this?? which method should b used for better computation??
I'm trying to find the regular parallelepiped with sides parallel to the coordinate axis inscribed in the ellipsoid x[2]/a[2] + y[2]/b[2] + z[2]/c[2] = 1 that has the largest volume. I've been trying the Lagrangian method: minimize f = (x)(y)(z), subject to the constraint (x[2]/a[2] + y[2]/b[2]...
Im not sure if this is the right place, but I have an optimization problem where I assume we are supposed to use the Lagraingian method:
Consider the labour supply problem for an individual over an entire year. Suppose the individuals utility is described by the function U = (C^0.5) x...