In this paper we study the approximation algorithms for a class of discrete quadratic optimization problems in the Hermitian complex form. A special case of the problem that we study corresponds to the max-3-cut model used in a recent paper of Goemans and Williamson. We first develop a closed-form formula to compute the probability of a complex-valued normally distributed bivariate random vector to be in a given angular region. This formula allows us to compute the expected value of a randomized (with a specific rounding rule) solution based on the optimal solution of the complex SDP relaxation problem. In particular, we study the limit of that model, in which the problem remains NP-hard. We show that if the objective is to maximize a positive semidefinite Hermitian form, then the randomization-rounding procedure guarantees a worst-case performance ratio of π/4 ≈ 0.7854, which is better than the ratio of 2/π ≈ 0.6366 for its counter-part in the real case due to Nesterov. Furthermore, if the objective matrix is real-valued positive semidefinite with non-positive off-diagonal elements, then the performance ratio improves to 0.9349.

This paper deals with the design of coded waveforms which optimize radar performances in the presence of colored Gaussian disturbance. We focus on the class of phase coded pulse trains and determine the radar code which approximately maximizes the detection performance under a similarity constraint with a pre-fixed radar code. This is tantamount to forcing a similarity between the ambiguity functions of the devised waveform and of the pulse train encoded with the pre-fixed sequence. We consider the cases of both continuous and finite phase alphabet, and formulate the code design in terms of a non-convex, NP-hard quadratic optimization problem. In order to approximate the optimal solutions, we propose techniques (with polynomial computational complexity) based on the method of Semidefinite Program (SDP) relaxation and randomization. Moreover, we also derive approximation bounds yielding a "measure of goodness" of the devised algorithms. At the analysis stage, we assess the performance of the new encoding techniques both in terms of detection performance and ambiguity function, under different choices for the similarity parameter. We also show that the new algorithms achieve an accurate approximation of the optimal solution with a modest number of randomizations.

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This paper studies the possibilities of the Linear Matrix Inequality (LMI) characterization of the matrix cones formed by nonnegative complex Hermitian quadratic functions over specific domains in the complex space. In its real case analog, such studies were conducted in Sturm and Zhang [11]. In this paper it is shown that stronger results can be obtained for the complex Hermitian case. In particular, we show that the matrix rank-one decomposition result of Sturm and Zhang [11] can be strengthened for the complex Hermitian matrices. As a consequence, it is possible to characterize several new matrix co-positive cones (over specific domains) by means of LMI. As examples of the potential application of the new rank-one decomposition result, we present an upper bound on the lowest rank among all the optimal solutions for a standard complex SDP problem, and offer alternative proofs for a result of Hausdorff [5] and a result of Brickman [3] on the joint numerical range.

Radar signal design in a spectrally crowded environment is currently a challenge due to the increasing requests for spectrum from both military sensing applications and civilian wireless services. The goal of this paper is to improve a previously devised algorithm for the synthesis of optimized radar waveforms fulfilling spectral compatibility with overlaid licensed radiators. The new technique achieves an enhanced spectral coexistence with the surrounding electromagnetic environment through a suitable modulation of the transmitted waveform energy, which was kept fixed at the maximum level in the previously devised algorithm. At the analysis stage, the waveform performance is studied in terms of trade-off among the achievable Signal to Interference Plus Noise Ratio (SINR), spectral shape, and the resulting Autocorrelation Function (ACF), also in situations where the previous technique cannot be applied.