The Complexity of Optimizing over a Simplex, Hypercube or Sphere: A Short Survey

We consider the computational complexity of optimizing various classes of continuous functions over a simplex, hypercube or sphere.These relatively simple optimization problems have many applications.We review known approximation results as well as negative (inapproximability) results from the recent literature.computational complexity;global optimization;linear and semidefinite programming;approximation algorithms

Ethylene and rooting of mung bean cuttings. The role of auxin induced ethylene synthesis and phase-dependent effects

We have re-examined the role of ethylene during rooting of mung bean cuttings. Cuttings were treated for 5 days with a low or a high concentration of NAA (naphthaleneacetic acid). During this 5 days period, we also applied STS (silverthiosulfate, an inhibitor of ethylene action) or ACC (1-aminocyclo-propane-l-carboxylic acid, a direct precursor of ethylene). At high NAA concentration, STS promoted and ACC inhibited rooting, respectively. At low NAA concentration, the effects were opposite, STS being inhibitory and ACC promotive. AVG (aminoethoxyvinylglycine, an inhibitor of ethylene synthesis) gave similar results as STS. Together, these data suggest supraoptimal and suboptimal ethylene levels in the tissue at high and low NAA concentration, respectively. We also examined whether the effect of ethylene varied during the successive phases of the rooting process. Thus, we gave 24 h pulses with either STS or ACC during the rooting treatment. During the first two days (0-48 h), ACC-pulses were promotive and STS-pulses inhibitory. Later on (48-168 h), the ACC-pulses were inhibitory and the STS-pulses promotive. Whether this effect was observed or not was dependent on the NAA concentration. These data indicate that ethylene promotes or inhibits rooting depending on the stage in the rooting process. When ACC was added only during the initial period, rooting was increased at all NAA concentrations in a NAA dose-response curve and the optimal NAA concentration remained the same. This suggests that ethylene renders more cells responsive to NAA

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) (2011), pp. 864-885], for the special case when the feasible set is the unit (hyper)sphere. The upper bound at level r of the hierarchy is defined as the minimal expected value of the polynomial over all probability distributions on the sphere, when the probability density function is a sum-of-squares polynomial of degree at most 2r with respect to the surface measure. We show that the exact rate of convergence is Theta(1/r^2), and explore the implications for the related rate of convergence for the generalized problem of moments on the sphere.Comment: 14 pages, 2 figure

Worst-case examples for Lasserre's measure--based hierarchy for polynomial optimization on the hypercube

We study the convergence rate of a hierarchy of upper bounds for polynomial optimization problems, proposed by Lasserre [SIAM J. Optim. 21(3) (2011), pp. 864-885], and a related hierarchy by De Klerk, Hess and Laurent [SIAM J. Optim. 27(1), (2017) pp. 347-367]. For polynomial optimization over the hypercube, we show a refined convergence analysis for the first hierarchy. We also show lower bounds on the convergence rate for both hierarchies on a class of examples. These lower bounds match the upper bounds and thus establish the true rate of convergence on these examples. Interestingly, these convergence rates are determined by the distribution of extremal zeroes of certain families of orthogonal polynomials.Comment: 17 pages, no figure

A note on the stability number of an orthogonality graph

We consider the orthogonality graph Omega(n) with 2^n vertices corresponding to the 0-1 n-vectors, two vertices adjacent if and only if the Hamming distance between them is n/2. We show that the stability number of Omega(16) is alpha(Omega(16))= 2304, thus proving a conjecture by Galliard. The main tool we employ is a recent semidefinite programming relaxation for minimal distance binary codes due to Schrijver. As well, we give a general condition for Delsarte bound on the (co)cliques in graphs of relations of association schemes to coincide with the ratio bound, and use it to show that for Omega(n) the latter two bounds are equal to 2^n/n.Comment: 10 pages, LaTeX, 1 figure, companion Matlab code. Misc. misprints fixed and references update

On Semidefinite Programming Relaxations of Association Schemes With Application to Combinatorial Optimization Problems

AMS classification: 90C22, 20Cxx, 70-08traveling salesman problem;maximum bisection;semidefinite programming;association schemes