New smooth gap function for box constrained variational inequalities

A new smooth gap function for the box constrained variational inequality problem （VIP） is proposed based on an integral global optimality condition. The smooth gap function is simple and has some good differentiable properties. The box constrained VIP can be reformulated as a differentiable optimization problem by the proposed smooth gap function. The conditions, under which any stationary point of the optimization problem is the solution to the box constrained VIP, are discussed. A simple frictional contact problem is analyzed to show the applications of the smooth gap function. Finally, the numerical experiments confirm the good theoretical properties of the method.

An ISGW Filtering Antenna with Spurious Modes and Surface Wave Suppression for Millimeter Wave Communications

In this paper,an integrated substrate gap waveguide(ISGW)filtering antenna is proposed at millimeter wave band,whose surface wave and spurious modes are simultaneously suppressed.A secondorder filtering response is obtained through a coupling feeding scheme using one uniform impedance resonator(UIR)and two stepped-impedance resonators(SIRs).To increase the stopband width of the antenna,the spurious modes are suppressed by selecting the appropriate sizes of the ISGW unit cell.Furthermore,the ISGW is implemented to improve the radiation performance of the antenna by alleviating the propagation of surface wave.And an equivalent circuit is investigated to reveal the working principle of ISGW.To demonstrate this methodology,an ISGW filtering antenna operating at a center frequency of 25 GHz is designed,fabricated,and measured.The results show that the antenna achieves a stopband width of 1.6f0(center frequency),an out-of-band suppression level of 21 dB,and a peak realized gain of 8.5 dBi.

Approximation Algorithms for Vertex Happiness

We investigate the maximum happy vertices(MHV)problem and its complement,the minimum unhappy vertices(MUHV)problem.In order to design better approximation algorithms,we introduce the supermodular and submodular multi-labeling(SUP-ML and SUB-ML)problems and show that MHV and MUHV are special cases of SUP-ML and SUB-ML,respectively,by rewriting the objective functions as set functions.The convex relaxation on the I ovasz extension,originally presented for the submodular multi-partitioning problem,can be extended for the SUB-ML problem,thereby proving that SUB-ML(SUP-ML,respectively)can be approximated within a factorof2-2/k(2/k,respectively),where k is the number of labels.These general results imply that MHV and MUHV can also be approximated within factors of 2/k and 2-2/k,respectively,using the same approximation algorithms.For the MUHV problem,we also show that it is approximation-equivalent to the hypergraph multiway cut problem;thus,MUHV is Unique Games-hard to achieve a(2-2/k-e)-approximation,for anyε>0.For the MHV problem,the 2/k-approximation improves the previous best approximation ratio max{1/k,1/(△+1/g(△)},where△is the maximum vertex degree of the input graph and g(△)=(√△+√△+1)2△>4△2.We also show that an existing LP relaxation for MHV is the same as the concave relaxation on the Lovasz extension for SUP-ML;we then prove an upper bound of 2/k on the integrality gap of this LP relaxation,which suggests that the 2/k-approximation is the best possible based on this LP relaxation.Lastly,we prove that it is Unique Games-hard to approximate the MHV problem within a factor of S2(log2 k/k).