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Sensitivity of Newton's method initial vlaue

The convergence of Newton's Method is sensitive to initial conditions (even if the initial point isn't obviously near a critical point) and can be difficult to predict. In many cases, applying the ...
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Black-box optimization of a symmetric function

Symmetries in optimization problems are considered as a curse rather than a benefit. They multiply the number of optimal points, so that algorithm that explore trees (e.g. branch-and-bound) have ...
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Constrain set of a constrained optimization problem

You are right that one reason to use a set $S$ smaller than all of $\mathbb R^n$ is to allow for your objective function (or for that matter the constraints) to be undefined outside $S$. Another ...
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How to transform this expression to a numerically stable form?

When computing $1-t$ for $t \to 1$ in finite-precision floating-point arithmetic, the result is exact per Sterbenz lemma. This leaves us with problems of subtractive cancellation. These can be ...
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What's the purpose of the KKT condition when first-order optimality condition exists?

... because it is easier to check whether the KKT conditions are satisfied compared to the first-order conditions. In fact, you only need some set of multipliers, plug it into the set of equations and ...
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Why do we linearize optimization problems?

We can’t solve high degree equation exactly. And even if we could, it would be very slow. The first order (linear) approximation steps iterate the linear equations till the precision is good enough. ...
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Constrain set of a constrained optimization problem

$S$ would be the domain of your design variables. Note that you can evaluate the objective function and other constraint functions outside $\{x\in\mathbb{R}^n:g(x)=0\}$. Moreover, some properties of ...
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Minimising inconclusive range for binary classification with two thresholds

You can linearize the problem and use a mixed integer linear programming solver, which will find a globally optimal solution without relying on an initial solution. Besides the decision variables $a$, ...
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Check the convexity of the upper bounding function of the subgradient method

You can rewrite your function as $$U(\alpha) = \frac{R^2}{2} \cdot \frac{1}{\sum_{i=1}^k \alpha_i} + \frac{M^2}{2} \cdot \frac{\|\alpha\|_2^2}{\sum_{i=1}^k \alpha_i},\: \alpha \in \mathbb{R}^k,$$ and ...
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What equation would express multiple values of x for only one value of n, within a threshold of requirements?

It seems likely that a square 2D array / matrix would be the best data structure to store your computed values in. Granted, such an array has $n^2$ entries, and you only need ${n\choose 2}$, but this ...
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How do I minimize sum of modules?

EDIT Inside a domain where all moduli contents have constant sign, the problem is a LP (Linear Program) and hence has an optimum on a vertex of the domain boundary. This implies that the global ...
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Notice that adding the same constant to each entry of $x$ doesn't change the objective function, which we denote by $$f_D(x) = \sum_{i < j} ((x_i - x_j) - D_{ij})^2.$$ Now, let $x^*$ minimise $... • 4,187 1 vote Prove that$\|∇f(x) − ∇f(y)\| ≤ M\|x − y\|.$Given$y$and$x$, define the function$g:[0,1] \rightarrow \mathbb{R}$, $$g(t) = (\nabla f(x) - \nabla f(y))^{T}(\nabla f(y+t(x-y)) - \nabla f(y)),$$ for all$t \in [0,1]$. Note that $$g'(t) = (\... • 832 1 vote Primal Interior-Point Methods You could apply a primal-dual approach or a primal approach to your problem. Both are valid. The primal approach to your problem is called the logarithmic barrier method. The primal-dual approach is ... • 832 1 vote Primal Interior-Point Methods If you go back to the original article by Fiacco & McCormick, see e.g. jstor, they do not introduce slack variables either. (Note: they work with 1/c_i(x) barrier instead of the logarithmic one, ... • 668 1 vote Accepted Can this formula be minimized in an exact way? r(x) = \sqrt{x^2 + (Kx + M)^2} An alternative to the other answer is to use calculus. Minimizing r(x) is the same as minimizing:$$R(x)=[r(x)]^2=x^2+(Kx+M)^2$$The first-order condition is$$R'(x)=2x+2K(Kx+M)=2(1+K^2)x+2KM=0$$... • 5,724 1 vote Can this formula be minimized in an exact way?$r(x) = \sqrt{x^2 + (Kx + M)^2}r^2=x^2+(Kx+M)^2= x^2+K^2x^2+2KMx+M^2r^2=x^2(1+K^2)+2KMx+M^2r^2/(1+K^2)=x^2+\frac{2KM}{1+K^2} x+\frac{M^2}{1+K^2}r^2/(1+K^2)=x^2+\frac{2KM}{1+K^2}x+ \frac{K^2M^2}{(1+K^2)^2}+\frac{M^2(1+K^2)...
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