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# Questions tagged [numerical-optimization]

Numerical methods for continuous optimization.

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### Convergence Rate of Projected Gradient Descent with Simplex Constraints

I'm trying to study the convergence rate, which is defined as $$\lim_{k \to \infty} \frac{f(x_{k+1}) - f(x_*)}{(f(x_k) - f(x_*))^p} = R$$ (where $x_k$ is the $k$-th iterate while $x_*$ is the ...
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### Proof of the convergence of a convex function with newton method

I need to prove the following: Assuming that $f$ is convex, under the assumption that $f$ is convex and $x^{(0)} \geq x_\star$, the algorithm always delivers a converging sequence: Theorem: Let's ...
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### Difference between augmented Lagrangian and penalty method

1.what is an auxiliary variable? 2.when should we use the augmented Lagrangian (AL) instead of Lagrangian multiplier? 3.when should we use the penalty function instead of augmented Lagrangian (AL) ? ...
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### Algorithm to find minimum of a multivariate function

Problem: Find (numerically) minimum of a function $f=f(x_1,...,x_M)$, where $M \in \mathbb{N}$ - a fixed number (often large), and $x_i \in [a,b],\forall i$. Function $f$ is complicated, and to ...
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### What point will ADMM converge to ? A feasible point or a stationary point or local optima or global optima?

In Boyd's great ADMM paper Section 3.2.1, ''Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers'', it says that as iteration index $k\to \infty$, ...
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### Globalized Newton method for minimizing a specific functional: Convergence?

I'm currently working on a generalized p-Laplace equation: \begin{align} \label{DP} \begin{cases} \text{div} (\sigma |\nabla u|^{p-2} \nabla u) = f &\quad \text{in } \Omega\\ u = g &\quad \...
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I am using gradient descent to solve the linear system $Ax=b$, where matrix $A$ is symmetric and positive definite. More precisely, I am attempting to solve the following quadratic program $$\text{... 0answers 10 views ### Reference on one iteration convergence of gradient projection algorithm (Bound constrained optimization)? So for a bound constrained optimization problem of minimizing a continuously differential function f(x) with x^* which is a non degenerate solution, I was interested in showing that if all bound ... 0answers 37 views ### Characterizing class of functions from their gradient descent trajectory Let f : \mathbb R^d \to \mathbb R be a function with at least one minimum. Suppose that f has the property that the gradient descent trajectory from any point of the function is a straight line to ... 0answers 15 views ### Superimpose an item of the chaotic sequence on vector In the article "Improved Chaotic Gravitational Search Algorithms for Global Optimization" on page 1223, in step 6 of the pseudo-code "Chaotic Local Search Algorithm", the phrase "Superimpose an item ... 0answers 27 views ### Is there efficient Surface Walking method for optimization problems with equality constraint? To my best knowledge, if we want to find the minimum of a function f defined on a d-dimension manifold M in \mathbb{R}^n, a.k.a an optimization problem with equality constraint, the most ... 0answers 12 views ### Roots of a n-variable non linear function with numerical methods Currently I am working with finding the solutions for the following problem: I have a unit sphere in which I have n points defined by their polar and azimuthal angles: \theta_n , \phi_n. I then do ... 1answer 20 views ### Prove that \xi_{k+1} = (-1)^{k+1}(\alpha_0 \times \alpha_1 \times \dots \times \alpha_k) is the (k+1) coefficient of p_k I was given the following question as part of a homework assignment. Any help would be greatly appreciated! The following image shows the steps of a preliminary version of the conjugate gradient ... 0answers 28 views ### Optimization with difference equation constraint I'm working on a problem where I have a (vector) linear recurrence relation of the form$$ a_{n+1} = \lambda \circ a_n+b_n $$I need to solve the following optimization problem:$$ \min\limits_{b_n}...
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I've found a variety of variations of writing Nesterov but I cannot understand why they cannot simply be expanded into a one liner. Here is one I found that can just be re-arranged, can someone ...
I am having trouble understanding it, first of all, what is x? Are x's coefficients of this polynomial we are looking for? This would mean that the polynomial is of degree $n-1$ because it has n ...