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The Dirac–Goodman–Pollack Conjecture
In one of their seminal articles on allowable sequences, Goodman and Pollack gave combinatorial generalizations for three problems in discrete...
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Symmetries in Polynomial Optimization
This chapter investigates how symmetries can be used to reduce the computational complexity in polynomial optimization problems. A focus will be...
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Edges and Cycles
In the previous chapters, we have been considering vertex colourings. Now we will consider edge colourings of a graph. We will say that two edges are...
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Structural Stability of the Financial Market Model: Continuity of Superhedging Price and Model Approximation
The present paper continues the topic of our recent paper in the same journal, aiming to show the role of structural stability in financial modeling....
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Generalizing Korchmáros—Mazzocca Arcs
In this paper, we generalize the so-called Korchmáros—Mazzocca arcs, that is, point sets of size q + t intersecting each line in 0, 2 or t points in...
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Convergence rates of RLT and Lasserre-type hierarchies for the generalized moment problem over the simplex and the sphere
We consider the generalized moment problem (GMP) over the simplex and the sphere. This is a rich setting and it contains NP-hard problems as special...
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Connecting optimization with spectral analysis of tri-diagonal matrices
We show that the global minimum (resp. maximum) of a continuous function on a compact set can be approximated from above (resp. from below) by...
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On the Structure of Pointsets with Many Collinear Triples
It is conjectured that if a finite set of points in the plane contains many collinear triples, then part of the set has a structure. We will show...
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On the effective Putinar’s Positivstellensatz and moment approximation
We analyse the representation of positive polynomials in terms of Sums of Squares. We provide a quantitative version of Putinar’s Positivstellensatz...
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Introduction
Optimization problems are as old as science itself, implicit in the behavior of humanity. These are defined by their specific objective functions,...
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Convergence analysis of a Lasserre hierarchy of upper bounds for polynomial minimization on the sphere
We study the convergence rate of a hierarchy of upper bounds for polynomial minimization problems, proposed by Lasserre (SIAM J Optim 21(3):864–885,...
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A Survey of Semidefinite Programming Approaches to the Generalized Problem of Moments and Their Error Analysis
The generalized problem of moments is a conic linear optimization problem over the convex cone of positive Borel measures with given support. It has...
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Positivity certificates and polynomial optimization on non-compact semialgebraic sets
In a first contribution, we revisit two certificates of positivity on (possibly non-compact) basic semialgebraic sets due to Putinar and Vasilescu (C...
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Certifying the Global Optimality of Quartic Minimization over the Sphere
The quartic minimization over the sphere is an NP-hard problem in the general case. There exist various methods for computing an approximate solution...
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On the freeness of Böröczky line arrangements
In the present note, we focus on the freeness and some combinatorial properties of line arrangements in the projective plane having only double and...
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Commentaries and Further Developments
In this chapter, we comment on and discuss some of the most important further developments obtained in the domains considered. More extensions are...