# Tag Info

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What about this: Step 1: For each edge $P_iP_j$, look at the two triangles that share it: $P_iP_jP_k$ and $P_lP_jP_i$. Step 2: Compute the counter-clockwise normal on each of the two triangle: $$\vec{n}_{ijk} = \frac{\vec{P_iP_j} \times \vec{P_jP_k}}{\|\vec{P_iP_j} \times \vec{P_jP_k}\|} \,\,\, \text{ and }\,\,\, \vec{n}_{lji} = \frac{\vec{P_lP_j} \times ... 0 I believe it can be done with a semidefinite program by adding a multiplicative slack variable. Basically,$$ \begin{array}{rcl} \min\limits_{A \in \mathbb{R}^{n \times n}, P\in \mathbb{R}^{n\times n}} &&f(A)\\ \text{st} && b I \preceq PA + A^TP \preceq a I\\ && P \succ 0 \end{array} $$Essentially, this is the Lyapunov stability ... 1 I have been thinking that it might be easier if one changes first the variables in the differential equation. That way one can bypass the function h(t) and deal with fewer functions. Since$$\hat{y}(t) = \hat{y}_{\theta}(t) = \hat{y}(t \,| \, \theta) = \theta \, \sqrt{2g} \, \sqrt{h(t)}$$change the dependent variable$$\hat{y} = \theta \, \sqrt{2g} \, \...

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I don't see how two equations give $z$ as output. Nevertheless, your sequence of computations looks reasonable, except I would combine steps one and two into: Simultaneously solve for $x$ and the sensitivities $z$. This is a extended ODE system that you could throw at a built-in MATLAB ODE solver: $$\begin{bmatrix} x'(t,\hat{\theta}) \\ z'(t,\hat{\theta}... 3 Despite my comment, I think you can find \tilde{D} that contains the noise term as well. You have this equation:$$-M^{T} \tilde{D} M \phi(t) = -M^{T} D M \phi(t) + W(t)$$Where W(t) is the noise term vector. So:$$-M^{T} (\tilde{D}-D) M \phi(t) = W(t)$$Take \mathcal{D} = \tilde{D} - D. Let's expand this equation:$$(M \phi(t))_{i} = \sum_{j=1}^{...

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A common approach to these kinds of problems is to sample the range of parameters (e.g. on a rectangular grid) and then fit a quadratic function to the points as a surrogate or "response surface" You then minimize the quadratic surrogate function. After doing one round of this, you can repeat the process using a finer grid around the minimum of the first ...

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From a pure convergence point of view, I believe the only thing that's necessary is to satisfy the convergence criteria from a globalization method such as a line-search or trust-region. This is generally required for convergence even if you used the exact gradient when determining your search direction. Meaning, BFGS will not generally converge by itself ...

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If your objective function is noisy, then it makes sense to use stochastic algorithms. I would take a look at James Spall's SPSA algorithm and variations. There are also algorithms that update (an approximation of) the Hessian. All of these algorithms take into account that whatever gradient you compute may be noisy, and consequently don't just blindly ...

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First of all, move the objective to the constraints by using the epigraph formulation max t subject to log_det(A) $\ge$ t plus your other constraints. Section 6.2.3 of the Mosek Modeling Cookbook https://docs.mosek.com/modeling-cookbook/sdo.html#semidefinite-modeling shows how to formulate log_det(A) $\ge$ t in terms of a combination of SDP and ...

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The determinant is the product of all eigenvalues $\lambda_i(A)$, so its logarithm is the sum of the logarithms of the eigenvalues. As a consequence, you can write the objective function as follows: $$\min_A \quad-\sum_i \log\lambda_i(A).$$ Now, assume you had an eigenvalue that tried to go to zero, then its logarithm would go to negative infinity and so ...

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