Sparse sum-of-squares certificates on finite abelian groups

Let G be a finite abelian group. This paper is concerned with nonnegative functions on G that are sparse with respect to the Fourier basis. We establish combinatorial conditions on subsets S and T of Fourier basis elements under which nonnegative functions with Fourier support S are sums of squares of functions with Fourier support T. Our combinatorial condition involves constructing a chordal cover of a graph related to G and S (the Cayley graph Cay($\hat{G}$,S)) with maximal cliques related to T. Our result relies on two main ingredients: the decomposition of sparse positive semidefinite matrices with a chordal sparsity pattern, as well as a simple but key observation exploiting the structure of the Fourier basis elements of G. We apply our general result to two examples. First, in the case where $G = \mathbb{Z}_2^n$, by constructing a particular chordal cover of the half-cube graph, we prove that any nonnegative quadratic form in n binary variables is a sum of squares of functions of degree at most $\lceil n/2 \rceil$, establishing a conjecture of Laurent. Second, we consider nonnegative functions of degree d on $\mathbb{Z}_N$ (when d divides N). By constructing a particular chordal cover of the d'th power of the N-cycle, we prove that any such function is a sum of squares of functions with at most $3d\log(N/d)$ nonzero Fourier coefficients. Dually this shows that a certain cyclic polytope in $\mathbb{R}^{2d}$ with N vertices can be expressed as a projection of a section of the cone of psd matrices of size $3d\log(N/d)$. Putting $N=d^2$ gives a family of polytopes $P_d \subset \mathbb{R}^{2d}$ with LP extension complexity $\text{xc}_{LP}(P_d) = \Omega(d^2)$ and SDP extension complexity $\text{xc}_{PSD}(P_d) = O(d\log(d))$. To the best of our knowledge, this is the first explicit family of polytopes in increasing dimensions where $\text{xc}_{PSD}(P_d) = o(\text{xc}_{LP}(P_d))$.Comment: 34 page

Bidimensional Inequalities with an Ordinal Variable

We investigate the normative foundations of two empirically implementable dominance criteria for comparing distributions of two attributes, where the first one is cardinal while the second is ordinal. The criteria we consider are Atkinson and Bourguignon\'s (1982) first quasi-ordering and a generalization of Bourguignon\'s (1989) ordered poverty gap criterion. In each case we specify the restrictions to be placed on the individual utility functions, which guarantee that all utility-inequality averse welfarist ethical observers will rank the distributions under comparison in the same way as the dominance criterion. We also identify the elementary inequality reducing transformations successive applications of which permit to derive the dominating distribution from the dominated one.Normative Analysis, Utilitarianism, Welfarism, Bidimensional Stochastic Dominance, Inequality Reducing Transformations

Restrukturiranje sustava fiskalnog izravnanja u Hrvatskoj

The aim of this paper is to propose a model of fiscal equalization in Croatia. This paper tests the hypothesis of a lack of effectiveness of the existing fiscal equalization model compared to a model that would be based on alleviating the difference in the potential to collect revenue from the personal income tax and surtax. Fiscal inequalities of local government units are determined first under the current equalization system by calculating the Gini coefficients and graphically presented with Lorenz curves. Thereafter, a distribution of equalization grants is simulated based on the new (proposed) model. The effectiveness of the proposed model in alleviating the fiscal inequalities is determined in relation to the effectiveness of the current equalization system. It was found that the model based on equalizing the difference in the capacity to collect revenue from the personal income tax and surtax alleviates inequalities in fiscal capacities of local government units much better than the existing system at the same cost. The main conclusion is that the fiscal equalization in Croatia should urgently be redesigned in order to improve efficiency and fairness, but also the transparency and credibility of the equalization system.Cilj rada je predložiti model fiskalnog izravnanja u Hrvatskoj. U radu se testira hipoteza o nedovoljnoj učinkovitosti postojećeg modela fiskalnog izravnanja u odnosu na model koji bi se temeljio na ublažavanju razlika u potencijalu prikupljanja prihoda od prireza i poreza na dohodak. Izračunom Ginijevih koeficijenata najprije se utvrđuju nejednakosti u raspodjeli fiskalnih kapaciteta po stanovniku jedinica lokalne samouprave u postojećem sustavu izravnanja koje se prikazuju i grafički Lorenzovim krivuljama. Potom se provodi simulacija raspodjele pomoći za fiskalno izravnanje na temelju novog (predloženog) modela te se testira njegova učinkovitost u ublažavanju fiskalnih nejednakosti. Utvrđeno je kako model utemeljen na ublažavanju razlika u kapacitetu prikupljanja prihoda od prireza i poreza na dohodak u znatno većoj mjeri od postojećeg sustava ublažava razlike u fiskalnim kapacitetima jedinica lokalne samouprave uz isti trošak. Zaključak je da fiskalno izravnanje u Hrvatskoj hitno treba restrukturirati kako bi se poboljšala učinkovitost i pravednost, ali i transparentnost i vjerodostojnost sustava izravnanja