Approximation bounds for quadratic maximization and max-cut problems with semidefinite programming relaxation

In this paper,we consider a class of quadratic maximization problems.For a subclass of the problems,we show that the SDP relaxation approach yields an approximation solution with the ratio is dependent on the data of the problem with α being a uniform lower bound.In light of this new bound,we show that the actual worst-case performance ratio of the SDP relaxation approach (with the triangle inequalities added) is at least α δd if every weight is strictly positive,where δd ＞ 0 is a constant depending on the problem dimension and data.     

Indestructibility and measurable cardinals with few and many measures

If κ < λ are such that κ is indestructibly supercompact and λ is measurable, then we show that both A = {δ < κ | δ is a measurable cardinal which is not a limit of measurable cardinals and δ carries the maximal number of normal measures} and B = {δ < κ | δ is a measurable cardinal which is not a limit of measurable cardinals and δ carries fewer than the maximal number of normal measures} are unbounded in κ. The two aforementioned phenomena, however, need not occur in a universe with an indestructibly supercompact cardinal and sufficiently few large cardinals. In particular, we show how to construct a model with an indestructibly supercompact cardinal κ in which if δ < κ is a measurable cardinal which is not a limit of measurable cardinals, then δ must carry fewer than the maximal number of normal measures. We also, however, show how to construct a model with an indestructibly supercompact cardinal κ in which if δ < κ is a measurable cardinal which is not a limit of measurable cardinals, then δ must carry the maximal number of normal measures. If we weaken the requirements on indestructibility, then this last result can be improved to obtain a model with an indestructibly supercompact cardinal κ in which every measurable cardinal δ < κ carries the maximal number of normal measures. A. W. Apter’s research was partially supported by PSC-CUNY grants and CUNY Collaborative Incentive grants. In addition, the author wishes to thank the referee, for helpful comments, corrections, and suggestions which have been incorporated into the current version of the paper.     

Geometric construction of association schemes from non-degenerate quadrics

LetF _q be a finite field withq elements, whereq is a power of an odd prime. In this paper, we assume that δ=0,1 or 2 and consider a projective spacePG(2ν+δ,F _q ), partitioned into an affine spaceAG(2ν+δ,F _q ) of dimension 2ν+δ and a hyperplaneℋ=PG(2ν+δ−1,F _q ) of dimension 2ν+δ−1 at infinity. The points of the hyperplaneℋ are next partitioned into three subsets. A pair of pointsa andb of the affine space is defined to belong to classi if the line meets the subseti of ℋ. Finally, we derive a family of three-class association schemes, and compute their parameters. This project is supported by the National Natural Science Foundation of China (No. 19571024).     

Gradient methods for multiple state optimal design problems

We optimise a distribution of two isotropic materials α I and β I (α < β) occupying the given body in R ^d . The optimality is described by an integral functional (cost) depending on temperatures u ₁, . . . , u _m of the body obtained for different source terms f ₁, . . . ,f _m with homogeneous Dirichlet boundary conditions. The relaxation of this optimal design problem with multiple state equations is needed, introducing the notion of composite materials as fine mixtures of different phases, mathematically described by the homogenisation theory. The necessary conditions of optimality are derived via the Gateaux derivative of the cost functional. Unfortunately, there could exist points in which necessary conditions of optimality do not give any information on the optimal design. In the case m < d we show that there exists an optimal design which is a rank-m sequential laminate with matrix material α I almost everywhere on Ω. Contrary to the optimality criteria method, which is commonly used for the numerical solution of optimal design problems (although it does not rely on a firm theory of convergence), this result enables us to effectively use classical gradient methods for minimising the cost functional.      

On dual minimum cost flow algorithms

We describe a new dual algorithm for the minimum cost flow problem. It can be regarded as a variation of the best known strongly polynomial minimum cost flow algorithm, due to Orlin. Indeed we obtain the same running time of O(m log m(m+n log n)), where n and m denote the number of vertices and the number of edges. However, in contrast to Orlin's algorithm we work directly with the capacitated network (rather than transforming it to a transshipment problem). Thus our algorithm is applicable to more general problems (like submodular flow) and is likely to be more efficient in practice.  Our algorithm can be interpreted as a cut cancelling algorithm, improving the best known strongly polynomial bound for this important class of algorithms by a factor of m. On the other hand, our algorithm can be considered as a variant of the dual network simplex algorithm. Although dual network simplex algorithms are reportedly quite efficient in practice, the best worst-case running time known so far exceeds the running time of our algorithm by a factor of n.     

Affine extractors over prime fields

An affine extractor is a map that is balanced on every affine subspace of large enough dimension. We construct an explicit affine extractor AE from \mathbbFⁿ \mathbb{F}^n to \mathbbF\mathbb{F}, \mathbbF\mathbb{F} a prime field, so that AE(x) is exponentially close to uniform when x is chosen uniformly at random from an arbitrary affine subspace of \mathbbFⁿ \mathbb{F}^n of dimension at least δn, 0<δ≤1 a constant. Previously, Bourgain constructed such affine extractors when the size of \mathbbF\mathbb{F} is two. Our construction is in the spirit of but different than Bourgain’s construction. This allows for simpler analysis and better quantitative results.     

Robust solutions to uncertain linear complementarity problems

In this paper,we adopt the robust optimization method to consider linear complementarity problems in which the data is not specified exactly or is uncertain,and it is only known to belong to a prescribed uncertainty set.We propose the notion of the p- robust counterpart and the p-robust solution of uncertain linear complementarity problems.We discuss uncertain linear complementarity problems with three different uncertainty sets,respectively,including an unknown-but-bounded uncertainty set,an ellipsoidal uncertainty set and an intersection-of-ellipsoids uncertainty set,and present some sufficient and necessary(or sufficient) conditions which p- robust solutions satisfy.Some special cases are investigated in this paper.     

When are static and adjustable robust optimization problems with constraint-wise uncertainty equivalent?

Adjustable robust optimization (ARO) generally produces better worst-case solutions than static robust optimization (RO). However, ARO is computationally more difficult than RO. In this paper, we provide conditions under which the worst-case objective values of ARO and RO problems are equal. We prove that when the uncertainty is constraint-wise, the problem is convex with respect to the adjustable variables and concave with respect to the uncertain parameters, the adjustable variables lie in a convex and compact set and the uncertainty set is convex and compact, then robust solutions are also optimal for the corresponding ARO problem. Furthermore, we prove that if some of the uncertain parameters are constraint-wise and the rest are not, then under a similar set of assumptions there is an optimal decision rule for the ARO problem that does not depend on the constraint-wise uncertain parameters. Also, we show for a class of problems that using affine decision rules that depend on all of the uncertain parameters yields the same optimal objective value as when the rules depend solely on the non-constraint-wise uncertain parameters. Finally, we illustrate the usefulness of these results by applying them to convex quadratic and conic quadratic problems.     

Affine Moser–Trudinger and Morrey–Sobolev inequalities

An affine Moser–Trudinger inequality, which is stronger than the Euclidean Moser–Trudinger inequality, is established. In this new affine analytic inequality an affine energy of the gradient replaces the standard L ⁿ energy of gradient. The geometric inequality at the core of the affine Moser–Trudinger inequality is a recently established affine isoperimetric inequality for convex bodies. Critical use is made of the solution to a normalized version of the L ⁿ Minkowski Problem. An affine Morrey–Sobolev inequality is also established, where the standard L ^p energy, with p > n, is replaced by the affine energy.     

Linear programming under uncertainty: A basic property of the optimal solution

Summary We show that if the two-stage linear programming problem under uncertainty has an optimal solution, then it has an optimal solution in which the column vectors corresponding to the positive first stage decision variables are linearly independent. This leads to the result that if an optimal solution exists, then there exists an optimal solution in which not more than m + ¯m of the first stage decision variables are positive. These results extend to the multi-stage case.This research was partially supported by the Office of Naval Research under Contract Nonr-222(83), the National Science Foundation under Contract GP-4593, the Army Research Office under Contract DA-31-124-ARO-D-331 and the University of California. Reproduction in whole or part is permitted for any purpose of the United States Government.     

Expanders obtained from affine transformations

A bipartite graphG=(U, V, E) is an (n, k, δ, α) expander if |U|=|V|=n, |E|≦kn, and for anyX⊆U with |X|≦αn, |Γ _G (X)|≧(1+δ(1−|X|/n)) |X|, whereΓ _G (X) is the set of nodes inV connected to nodes inX with edges inE. We show, using relatively elementary analysis in linear algebra, that the problem of estimating the coefficientδ of a bipartite graph is reduced to that of estimating the second largest eigenvalue of a matrix related to the graph. In particular, we consider the case where the bipartite graphs are defined from affine transformations, and obtain some general results on estimating the eigenvalues of the matrix by using the discrete Fourier transform. These results are then used to estimate the expanding coefficients of bipartite graphs obtained from two-dimensional affine transformations and those obtained from one-dimensional ones.     

A note on issues of over-conservatism in robust optimization with cost uncertainty

We identify and discuss issues of hidden over-conservatism in robust linear optimization, when the uncertainty set is polyhedral with a budget of uncertainty constraint. The decision-maker selects the budget of uncertainty to reflect his degree of risk aversion, i.e. the maximum number of uncertain parameters that can take their worst-case value. In the first setting, the cost coefficients of the linear programming problem are uncertain, as is the case in portfolio management with random stock returns. We provide an example where, for moderate values of the budget, the optimal solution becomes independent of the nominal values of the parameters, i.e. is completely disconnected from its nominal counterpart, and discuss why this happens. The second setting focusses on linear optimization with uncertain upper bounds on the decision variables, which has applications in revenue management with uncertain demand and can be rewritten as a piecewise linear problem with cost uncertainty. We show in an example that it is possible to have more demand parameters equal their worst-case value than what is allowed by the budget of uncertainty, although the robust formulation is correct. We explain this apparent paradox.     

Friabilité des valeurs d’un polynôme

Let F(X) be an absolutely irreducible polynomial in \mathbbZ [X₁,..., X_n]{\mathbb{Z} [X_{1},\dots, X_{n}]}, with degree d. We prove that, for any δ < 4/3, for any sufficiently large x, there exists a positive density of integral n-tuples m = (m ₁, . . . , m _n ) in the hypercube max |m _i | ≤ x such that every prime divisor of F(m) is smaller than x ^d–δ . This result is improved when F satisfies some geometrical hypotheses.     

A construction of polynomial lattice rules with small gain coefficients

In this paper we construct polynomial lattice rules which have, in some sense, small gain coefficients using a component-by-component approach. The gain coefficients, as introduced by Owen, indicate to what degree the method improves upon Monte Carlo. We show that the variance of an estimator based on a scrambled polynomial lattice rule constructed component-by-component decays at a rate of N ^{−(2α+1)+δ} , for all δ > 0, assuming that the function under consideration has bounded variation of order α for some 0 < α ≤ 1, and where N denotes the number of quadrature points. An analogous result is obtained for Korobov polynomial lattice rules. It is also established that these rules are almost optimal for the function space considered in this paper. Furthermore, we discuss the implementation of the component-by-component approach and show how to reduce the computational cost associated with it. Finally, we present numerical results comparing scrambled polynomial lattice rules and scrambled digital nets.     

Reduced Vertex Set Result for Interval Semidefinite Optimization Problems

In this paper we propose a reduced vertex result for the robust solution of uncertain semidefinite optimization problems subject to interval uncertainty. If the number of decision variables is m and the size of the coefficient matrices in the linear matrix inequality constraints is n×n, a direct vertex approach would require satisfaction of 2 ^{n(m+1)(n+1)/2} vertex constraints: a huge number, even for small values of n and m. The conditions derived here are instead based on the introduction of m slack variables and a subset of vertex coefficient matrices of cardinality 2 ⁿ⁻¹, thus reducing the problem to a practically manageable size, at least for small n. A similar size reduction is also obtained for a class of problems with affinely dependent interval uncertainty. This work is supported by MIUR under the FIRB project “Learning, Randomization and Guaranteed Predictive Inference for Complex Uncertain Systems,” and by CNR RSTL funds.     

On the Lipschitzian properties of polyhedral multifunctions

In this paper, we show that for a polyhedral multifunctionF:R ⁿ →R ^m with convex range, the inverse functionF ⁻¹ is locally lower Lipschitzian at every point of the range ofF (equivalently Lipschitzian on the range ofF) if and only if the functionF is open. As a consequence, we show that for a piecewise affine functionf:R ⁿ →R ⁿ ,f is surjective andf ⁻¹ is Lipschitzian if and only iff is coherently oriented. An application, via Robinson's normal map formulation, leads to the following result in the context of affine variational inequalities: the solution mapping (as a function of the data vector) is nonempty-valued and Lipschitzian on the entire space if and only if the solution mapping is single-valued. This extends a recent result of Murthy, Parthasarathy and Sabatini, proved in the setting of linear complementarity problems. Research supported by the National Science Foundation Grant CCR-9307685.     

On the number of Galois points for a plane curve in positive characteristic,III

We consider the following problem: For a smooth plane curve C of degree d ≥ 4 in characteristic p > 0, determine the number δ(C) of inner Galois points with respect to C. This problem seems to be open in the case where d ≡ 1 mod p and C is not a Fermat curve F(p ^e  + 1) of degree p ^e  + 1. When p ≠ 2, we completely determine δ(C). If p = 2 (and C is in the open case), then we prove that δ(C) = 0, 1 or d and δ(C) = d only if d−1 is a power of 2, and give an example with δ(C) = d when d = 5. As an application, we characterize a smooth plane curve having both inner and outer Galois points. On the other hand, for Klein quartic curve with suitable coordinates in characteristic two, we prove that the set of outer Galois points coincides with the one of \mathbbF₂{\mathbb{F}_{2}} -rational points in \mathbbP²{\mathbb{P}^{2}}.     

An Integral Simplex Algorithm for Solving Combinatorial Optimization Problems

In this paper a local integral simplex algorithm will be described which, starting with the initial tableau of a set partitioning problem, makes pivots using the pivot on one rule until no more such pivots are possible because a local optimum has been found. If the local optimum is also a global optimum the process stops. Otherwise, a global integral simplex algorithm creates and solves the problems in a search tree consisting of a polynomial number of subproblems, subproblems of subproblems, etc. The solution to at least one of these subproblems is guaranteed to be an optimal solution to the original problem. If that solution has a bounded objective then it is an optimal set partitioning solution of the original problem, but if it has an unbounded objective then the original problem has no feasible solution. It will be shown that the total number of pivots required for the global integral simplex method to solve a set partitioning problem having m rows, where m is an arbitrary but fixed positive integer, is bounded by a polynomial function of n.A method for programming the algorithms in this paper to run on parallel computers is discussed briefly.     

Constructions and bounds on linear error-block codes

We obtain new bounds on the parameters and we give new constructions of linear error-block codes. We obtain a Gilbert–Varshamov type construction. Using our bounds and constructions we obtain some infinite families of optimal linear error-block codes over . We also study the asymptotic of linear error-block codes. We define the real valued function α _q,m,a (δ), which is an analog of the important real valued function α _q (δ) in the asymptotic theory of classical linear error-correcting codes. We obtain both Gilbert–Varshamov and algebraic geometry type lower bounds on α _q,m,a (δ). We compare these lower bounds in graphs.      

The Number of Generalized Balanced Lines

Let S be a set of r red points and b=r+2δ blue points in general position in the plane, with δ≥0. A line ℓ determined by them is balanced if in each open half-plane bounded by ℓ the difference between the number of blue points and red points is δ. We show that every set S as above has at least r balanced lines. The proof is a refinement of the ideas and techniques of Pach and Pinchasi (Discrete Comput. Geom. 25:611–628, 2001), where the result for δ=0 was proven, and introduces a new technique: sliding rotations.