Interior Point Methods for Nonlinear Optimization

Pólik, Imre; Terlaky, Tamás

doi:10.1007/978-3-642-11339-0_4

Cited by 296 publications

(171 citation statements)

References 87 publications

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“…9 and 10(a) in this paper's Appendix. Coding used the Opt matlab toolbox [2], [3] interfaced to the SeDuMi numerical solver [4], [5]. Convergence, in under a second, used three LP/SOCP cycles.…”

Section: B Ifir Filtersmentioning

confidence: 99%

Nonseparable Nth-band filters as overlapping-subarray tapers

Coleman

2011

2011 IEEE RadarCon (RADAR)

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Abstract-In a receive array "overlapped subarrays" refers to an array output formed as a weighted sum of subarray outputs themselves formed as identically weighted combinations of the outputs of overlapping element subsets. The system array factor becomes a product of two array factors with different periods, one associated with each sum, just as the frequency response of a two-stage IFIR filter in a DSP system is the product of two frequency responses with different periods. It is well known that an Nth-band filter can be the most efficient choice for an IFIR filter's first stage, but the corresponding array idea appears unknown.Design examples apply modern optimization to nonseparable tapers, with three of four examples featuring Nth-band subarray tapers. Whether an Nth-band approach is best is not settled here and is properly context dependent: designs should be carried out both ways and compared. Further, a very small subarray taper can easily be Nth-band entirely by accident, so designers should be aware of the associated array-factor features that necessarily result and cannot be optimized away. I. BACKGROUND AND INTRODUCTION A. Review of overlapped subarraysIn the overlapped-subarray receive-array architecture of Fig. 1, identically weighted sums of (typically few) signals from overlapping element subsets become subarray outputs. The array output is a final weighted sum of those (typically many) subarray outputs. While design of the detailed Fig. 2 example used an optimal approach discussed below, a simple two-step design approach is more typical.The first step designs the final array factor of Fig. 2(a), the Fourier transform of the final taper (summing weights) of Fig. 2(e), to fix the mainlobe shape. (More below about its apparently unreasonable width.) The small period of that final array factor replicates the mainlobe to create the two grating lobes of Fig. 2(a). The second step designs the subarray array factor, the Fourier transform of the subarray taper of Fig. 2(d), to suppress those grating lobes in the product of the final and subarray array factors, the system array factor of Fig. 2(b).Instead of one final sum there are typically several in parallel, creating several array outputs or "beams" . Their final array factors likely have narrower main beams than in Fig. 2(a) and are often identical except for horizontal translation. Five narrow mainlobes of five output beams might, for example, appear where the five peaks of the wide example mainlobe are in Fig. 2(a). Here that wide mainlobe stands in informally for such a cluster of five narrow mainlobes, with the subarray array factor suppressing the grating lobes of each. (More precise illustration of a beam cluster is deferred to the 2D examples of Figs. 4 through 7.) Element-output phase shifts not shown in Fig. 1 (but see vector kc in the Appendix's detailed formulation) can steer the cluster as a unit by translating all array factors in lockstep. B. IFIR filtersIt is well known that when subarraying is not used the mathematics of a uniform li...

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Section: B Ifir Filtersmentioning

confidence: 99%

Nonseparable Nth-band filters as overlapping-subarray tapers

Coleman

2011

2011 IEEE RadarCon (RADAR)

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“…The complexity of solving an SDP problem is O n 0.5 sdp m sdp n 3 sdp + m 2 sdp n 2 sdp + m 3 sdp log(1/ ) according to [36]. Here, m sdp denotes the number of semidefinite cone constraints, n sdp denotes the dimension of the semidefinite cone, and is the accuracy of solving the SDP.…”

Section: Remarkmentioning

confidence: 99%

“…Here, m sdp denotes the number of semidefinite cone constraints, n sdp denotes the dimension of the semidefinite cone, and is the accuracy of solving the SDP. Comparing the SDP of (34) with the standard form in [36], we have m sdp = K + 4 and n sdp = N + 1. As a result, the computational complexity of the proposed scheme is…”

Section: Remarkmentioning

confidence: 99%

Robust beamforming and cooperative jamming for secure transmission in DF relay systems

Fei

2016

J Wireless Com Network

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In this paper, we study robust joint beamforming and cooperative jamming (CJ) in a secure decode-and-forward (DF) relay system in the presence of multiple eavesdroppers, in which a multi-antenna DF relay employs transmit beamforming to help the source deliver information to the destination and simultaneously generates Gaussian artificial noise to confuse these eavesdroppers. We assume that all the channel state information (CSI) is imperfectly known at the relay subject to norm-bounded CSI errors. Our objective is to maximize the achievable secrecy rate at the destination by jointly optimizing the transmit beamforming vector of information signals and the covariance matrix of jamming signals at the relay subject to its total transmit power constraint. Although this problem is non-convex and in general difficult to be solved, we obtain its optimal solution via the technique of semi-definite relaxation (SDR) together with a one-dimensional (1-D) search. Specifically, we prove that the SDR is tight for the problem of our interest. Finally, numerical results are provided to validate our proposed scheme.

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“…In the case when the EVM constraint is active, the worst-case complexity of the new approach is the same as that of SOCP WANG. See [19] for more details about complexity issues. We see that the complexity of a convex-based approach is higher than that of RCF.…”

Section: Complexity Analysismentioning

confidence: 99%

Peak reduction in OFDM using second-order cone programming relaxation

Beko

Dinis

Šendelj

2014

EURASIP J. Adv. Signal Process.

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In this paper, we address the problem of peak-to-average power ratio (PAPR) reduction in orthogonal frequency-division multiplexing (OFDM) systems. We formulate the problem as an error vector magnitude (EVM) optimization task with constraints on PAPR and free carrier power overhead (FCPO). This problem, which is known to be NP hard, is shown to be approximated by a second-order cone programming (SOCP) problem using a sequential convex programming approach, making it much easier to handle. This approach can be extended to the more general problem when PAPR, EVM, and FCPO are constrained simultaneously. Our performance results show the effectiveness of the proposed approach, which allows good performance with lower computational complexity and infeasibility rate than state-of-the-art PAPR-reduction convex approaches. Moreover, in the case when all the three system parameters are constrained simultaneously, the proposed approach outperforms the convex approaches in terms of infeasibility rate, PAPR, and bit error rate (BER) performances.

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Interior Point Methods for Nonlinear Optimization

Cited by 296 publications

References 87 publications

Nonseparable Nth-band filters as overlapping-subarray tapers

Nonseparable Nth-band filters as overlapping-subarray tapers

Robust beamforming and cooperative jamming for secure transmission in DF relay systems

Peak reduction in OFDM using second-order cone programming relaxation

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