# Questions tagged [numerical-optimization]

Numerical methods for continuous optimization.

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### Cost Optimization Given Constraints

Balance = 1,360 USD Cost = (0.0001% * Purchase Qty * Purchase Qty) + (1% * Purchase Qty) Problem: Find maximum Purchase Qty while keeping Cost below Balance. Is there a closed form solution to this?
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### Lagrangian Duality problem

Suppose that $\lambda, N_0 \in \mathbb{R}$, $\boldsymbol{X}_{0} = (X_2',...,X_{m+1}')' \in \mathbb{R}^{m \times n} , X_1 \in \mathbb{R}^n$ is fixed, $\iota \in \mathbb{R}^n$ is the vector of ones, ...
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### What numerical integration method can solve a integral over a function that depends on a differential equation without an analytic solution?

Suppose we have a differential equation of the form $\frac{dy}{dx}=f_{p_x,p_y}(y,x)$. After solving this differential equation for particular values of $p_x$ and $p_y$, we obtain $y_{p_x,p_y}(x)$. ...
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### How to solve this optimization problem? If there is no close-form solution, an approximate numerical results also helps.

I am stuck in an optimization problem. I only know Lagrange multipliers. However, it looks useless for such a complicated problem. $l, k, f$ are variables. Based on some domain knowledge, the range of ...
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### Physical interpretation of gradient descent with momentum

This is the Stochastic Gradient Descent with Momentum (SGD-M) equivalent question of the Batch Gradient Descent masterly answered here. In most of the literature where SGD-M is treated, the physical ...
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I encountered the following problem: Given the function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ given by $f(x,y) = y^2-x^2y$, I will omit the calculation of the gradient and Hessian but one can see the ...
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### Minimizing a function $f$ of the form $f(x)=\prod_if(x_i)$, is it possible to ensure $x_1,\ldots,x_j$ minimizes $\prod_i^jf(x_i)$?

Let $\varphi:\mathbb R\to\mathbb R$ with $$\varphi(x)\xrightarrow{|x|\to\infty}0,$$ $d\in\mathbb N$ and $$f_d(h):=\prod_{i=1}^d\sum_{k\in\mathbb Z}\varphi(h_i-k)\;\;\;\text{for }x\in\mathbb R^d.$$ ...
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### Minimize the variance of a nonnegative linear least-squares problem

My question is related to the Question: Minimize Variance of a Linear Function But with an additional contraint. Given a Matrix $A \in \mathbb{R}_{n\times m}$ and vector $\vec{b} \in \mathbb{R}_{m}$ . ...
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### Derivation of backwords SOR iterative method using matrices in numerical analysis

Cheers, while studying numerical analysis, I came across the following statement: SOR can be written as: $x^{(k+1)} = (I - \omega L)^{-1}[(1-\omega)I + \omega U] x^{(k)} + \omega(I-\omega L) ^{-1} c$ ...
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### Is there a reason why you would build a non-separable problem from a separable one? i.e. Rastrigin function

I'm reading on the Rastrigin function and I read that this function is additively decomposable, my understanding of this is that I could optimize this function by adding up its n independent 1-D ...
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### Are there any ODEs like these?

I have been working on using automatic differentiation to compute the sensitivities of systems of ordinary differential equations of the form \begin{equation} \frac{d\mathbf{u}(t)}{dt} = F(\mathbf{u}(...
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### Constrained optimization: When are the Lagrange multipliers bounded?

In some texts I have seen arguments for the fact that the Lagrange multipliers of a constrained optimization problem remain bounded. Are there general conditions for that fact? In particular, Let a ...
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