## Which non convex optimization algorithms guarantee a global optima?

Most non-convex optimization algorithms I have come across so far rely basically on random restart to find a better solution. e.g. Genetic Algorithm, Simulated Annealing, Metropolis Hastings Monte Carlo. Even in Stochastic Gradient Descent we can think of training each batch as a random perturbation.

There are some non-statistical algorithms in Dynamic Programming and Backtracking which can guarantee global optima, but I havent found any literature that works on noisy data.

What are the strategies which algorithms use (other than random restart) to get out of the local optima and search for the global optima for noisy data?

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## convex geometry – Cone related inequalities

I was working with some convex cones and came up with this inequality from the geometry:
Let $$Csubset R^p$$ be a closed convex cone and $$C^*$$ be its dual cone. Then for any $$thetain C$$ and $$gin R^p$$, the following inequality hold
$$|Pi_{C^*}(theta + g)| le |Pi_{C^*}(g)|,$$
where $$Pi_{C^*}$$ is the orthogonal projection function onto the convex cone $$C^*$$.

This inequality is straightforward from the geometry. I’d like to learn more about this type of inequalities. However, I didn’t find any reference on it. Can anyone direct me to the right reference on this type of inequalities about cones (or convexity)?

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## algorithms – Intersection test for convex polygons

I need to check if two convex polygons intersect of not, in a efficient way. The intersection itself is not required.

I know the method of separating axis, but it takes $$NM$$ tests for polygons of sizes $$N$$ and $$M$$.

I also know the efficient methods based on rotating calipers (Toussaint), “edge chasing” (O’Rourke), sweepline (Shamos & Hoey), Voronoi diagrams (Yang & al.), which all take $$N+M$$ tests.

Unfortunately, all these are designed with the idea that the polygons do intersect and the focus is on computing the intersection. For this reason, they don’t really describe what happens when the polygons in fact do not intersect and how to cope.

So what I am after is a simple procedure not taking more than $$N+M$$ tests and able to tell if the polygons intersect of not (or if one is wholly inside the other).

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## pr.probability – Lower bound on L2 norm of a strongly convex function

Let $$f(x):(0, 1) rightarrow R$$ be an $$m$$-strongly convex function and $$mu$$ be probability measure on $$(0,1).$$ For any $$t<1$$, the gaol is to find lowerbound on $$int_{0}^t f^2(x) dmu(x)$$ in terms of $$t$$, $$m$$, and $$mu$$ (and nothing else). We currently have the following bound
$$int_{0}^t f^2(x) dx ge frac{ m^2 t^4}{36} mu(0,t).$$
We do not know if our bound is tight. Moreover, our proof is really long and messy. A clean/simple proof of such an elementary result would be helpful.

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## convex optimization – Quasi-Concavity of Minimum of Function

Consider a differentiable function $$F(x,y,z)$$ defined on $$(0,1)times(0,1)times(0,1)$$, which is increasing and quasi-concave in (x,y). That is, the partial derivatives of $$F$$ with respect to $$x$$, $$y$$ are nonnegative, and we have
$$F(lambda x+(1-lambda)x’, lambda y+(1-lambda)y’,z)geq min{F(x,y,z),F(x’,y’,z)}$$

for any $$(x,y)$$, $$(x’,y’)$$ and $$lambdain(0,1)$$. I want to check whether we can prove
$$minlimits_{zin(lambda x+(1-lambda)x’,1)}F(lambda x+(1-lambda)x’, lambda y+(1-lambda)y’,z)geqmin{minlimits_{zin(x,1)}F(x,y,z),minlimits_{zin(x’,1)}F(x’,y’,z)}$$
or not.

I’ll appreciate any hints for this problem. Feel free to add any assumptions or corrections if necessary. Thanks!!

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## On Reflection Properties of Convex Regions

It is well known that any ray of light passing thru a focus of an ellipse will pass thru the other focus after a single reflection from the ellipse boundary. If A and B are the foci of an ellipse, this property of rays holds both ways (those passing thru A meet at B and vice versa).

1. Is there a closed convex region C with the property: there exists a pair of points A and B within C such that all rays thru A will reflect once on C and pass thru B but not all rays thru B will pass thru A after one reflection from C?

2. Is there a closed convex region C such that: there is a pair of points A and B in the interior such that all rays thru A pass thru B after exactly 2 reflections from C?
Note: This question can have ‘one-way’ (convergence only of rays thru A at B) and ‘both-ways’ variants.

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## Convex combination of (bounded) random variables can lead to a strictly larger norm?

Consider an arbitrary non-negative random variable $$Z in L^infty$$ such that $$E(Z)=1$$ and $$lVert Z rVert_{2}=2$$.

Does there exist a non-negative random variable $$Y in L^infty$$ such that $$E(Y)=1$$ and $$lVert lambda Y + (1-lambda)Z rVert_{2} > 2$$ for all $$lambda in (0,1)$$?

My thinking is yes, because the $$L^2$$-norm of $$Y$$ can be as large as we like. But I am finding it difficult to prove. My idea is that since $$Z$$ is in $$L^infty$$, you can construct $$Y$$ so that: $$Y>Z$$ for large values of $$Z$$, $$Y for small values of $$Z$$, and otherwise $$Y=Z$$ (making sure $$mathbb{E}(Y)=1$$). But writing this down formally is tricky. Is this the right way to go about it? Perhaps there is a way of proceeding via contradiction.

Extension: What happens if $$Z in L^2$$ (and so not necessarily bounded)?

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## mg.metric geometry – Getting more out of Minkowski’s convex body theorem in the case of non-convex bodies

Problem. In number theory one generally proves the finiteness of the Picard group of a number field using Minkowski’s convex body theorem. The actual body $$S_p$$ of interest in the proof, depending on some parameter $$p$$, is symmetric but not convex. We proceed by isolating the largest symmetric convex subset $$C$$ of $$S_p$$. The result is a worse bound on the $$p$$ for which we can find non-zero lattice points in $$S_p$$, than if we were allowed to apply Minkowski to $$S_p$$ instead of $$C$$.

Question. Are there general techniques to get better results than such a ‘wasteful’ application of Minkowski’s theorem for symmetric but non-convex bodies? With general I mean applicable to some lattices and bodies, but not necessarily to the above problem.

The case I am particularly interested in is asymptotic: I have for every dimension $$d$$ a lattice $$Lambda_d$$ and a symmetric body $$S_d$$ such that $$log(text{vol}(S_d)/text{det}(Lambda_d))=Theta(d^2)$$ and want to find a $$d$$ and a non-zero lattice point in $$Lambda_dcap S_d$$.

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## computational geometry – Concentric convex hulls

Given N points in a 2D plane, if we start at a given point and start including points in a set ordered by their distance from the starting point. After including every point, we check if there is a convex hull possible, we move all these points to a visited set and continue. Every time we determine a convex hull, we only move the points to the visited set if the so constructed convex hull contains all the points from the visited set.
How can we do count such convex hulls in as efficient manner as possible?

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## convex geometry – Euclidean volume of symmetric matrices in operator norm

This is a nearly identical question to Euclidean volume of the unit ball of matrices under the matrix norm except in the symmetric case.

Let $$mathrm{Sym}_{n times n}(mathbb{R})$$ be the space of real-valued $$n times n$$ symmetric matrices. Let $$phi : mathbb{R}^{n(n+1)/2} mapsto mathrm{Sym}_{n times n}(mathbb{R})$$ embed $$mathbb{R}^{n(n+1)/2}$$ into $$mathrm{Sym}_{n times n}(mathbb{R})$$.
Consider the set $$H_n = { v in mathbb{R}^{n(n+1)/2} : | phi(v) | leq 1 }$$, where $$|M| = max_{|x|=1} |Mx|$$ is the $$ell_2 mapsto ell_2$$ operator norm.

What is the formula for $$mathrm{Vol}(H_n)$$, where $$mathrm{Vol}(cdot)$$ is the Lebesgue measure on $$mathrm{R}^{n(n+1)/2}$$?

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