# 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.

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1. ### B Constrained Optimization with the KKT Approach

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...
2. F

### Stationary points classification using definiteness of the Lagrangian

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...
3. ### Optimization Problem - Dynamic Programming

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...
4. ### Maximizing the volume of a cylinder

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...
5. ### I Total Derivative of a Constrained System

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...
6. ### Is there a worked-out example of L-BFGS / L-BFGS-B?

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...
7. ### I Dot product constrained optimization

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)...
8. ### Constrained Optimization

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...
9. ### Nonlinear constrained optimization - how?

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...
10. ### Grouping constrained optimization

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...
11. ### MHB Molly's question at Yahoo Questions regarding constrained optimization

Here is the question: I have posted a link there to this thread so the OP can view my work.
12. ### Constrained Optimization Proof

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...
13. ### Constrained Optimization using Lagrange multipliers with Commerce applications

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...
14. ### Variable reduction on constrained optimization techniques

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...
15. ### How can I find the optimal vector x for a constrained optimization problem?

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...
16. ### Constrained Optimization via Lagrange Multipliers

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...
17. ### Multivariable Constrained Optimization

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??
18. ### Constrained optimization troubles

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]...
19. ### Lagraingian constrained optimization problem

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...