# Questions tagged [nonlinear-optimization]

A non-linear optimization problem includes an objective function (to be minimized or maximized) and some number of equality and/or inequality constraints where the objective or some of the constraints are non-linear. Use this tag for questions related to the theory of solving such problems or for trying to solve particular problems.

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### Non-linear programming problem with exponential objective function.

If I have an exponential objective function, but the problem is still convex. What is the best solver to use for my model?
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### Geometric interpretation of maximizing utility functions

So this was the problem I was trying to solve, and I got stuck in part (b). When you total differentiate $U(x,y)=0$, we can see that $Uxdx+Uydy=0$, which means $dy/dx=-Ux/Uy$ However, I am struggling ...
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### condition for convex function to have unique minimum value

The question is related to Convex function with unique critical point is coercive Assume we have a function $f: \mathbb{R}^{n} \to \mathbb{R}$ and we know that $f$ is just convex, not strictly nor ...
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### A Number of Jacobian entries

I'm trying to understand an optimization problem from IPOPT package. ...
1 vote
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1 vote
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### Implementing gradient optimization methods for non-linear functionals

I'm trying to write a code solving the Dirichlet boundary problem for $p$-Laplacian on an arbitrary planar domain $\Omega$: \begin{cases} -\Delta_p (u) = f \text{ on } \Omega, \\ u\big|_{\partial \...
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### Why do we linearize optimization problems?

I am currently doing research on the calibration of the robots' geometry, which is a standard and well-studied topic. In fact, it can be formulated as a nonlinear non-convex optimization problem: ...
1 vote
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### Question About Fritz John Theorem and Slater Constraint Qualification

Background Information I am studying constraint qualifications. Here are two theorems leading to my question: Theorem 1$\space\space\space\space$ [Fritz John Theorem] Suppose that $f, g_1, \dots, g_k$...
1 vote
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### Proof of Second Order Conditions for Maximization Problem with Two Variables and One Inequality Constraint

Background Information I am studying the theorem of S.O.C.s for a constrained maximization problem. Here is the theorem on the book but lack of proof: Theorem$\space\space\space\space$ Let $f$, $g_1$,...
1 vote
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### Prove that if the minimizer $\lambda$ of this parabola is feasible in ($*$), we must have $\alpha \in (0, 1/2)$.

The criterion of sufficient decrease (Armijo condition) requires $\lambda \in \mathbb{R}$ such that: \begin{equation} \varphi(\lambda) = f(x + \lambda d) < f(x) + \alpha \lambda \nabla^T f(x) d = \...
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### Constrain set of a constrained optimization problem

In the lecture notes I have, the constrain set in the equality case is defined as $S \cap \{x\in \mathbb{R}^n : g(x)=0\}$ where $S \subseteq \mathbb{R}^n$ open. The inequality and equality-inequality ...
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### Weighted least squares closed form solution

Warning - it's been awhile since I've dived into matrix+optimization math, so bear with me. I'm trying to determine the center of rotation of a single 3D link (think a rod attached to a ball joint) ...
1 vote
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### About Implicit Function Theorem and Lagrange Multipliers

I am studying the meaning of the multiplier in Lagrange Multiplier Method and got the following question. I tried it myself, but I am not sure if it is correct. I would really appreciate it if someone ...
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### The Intuition of Non Degenerate Constraint Qualification and Its Application When The Number of Binding Constraints Is More Than That of Variables

I am studying constrained optimization, and I am a bit confused by the Non Degenerate Constraint Qualification (NDCQ), its intuition and application. First, I would like to put the theorem of ...