Browsing by Subject "Semidefinite programming"

Now showing 1 - 7 of 7

Open Access
FIR filter design by convex optimization using directed iterative rank refinement algorithm
(Institute of Electrical and Electronics Engineers Inc., 2016) Dedeoğlu, M.; Alp, Y. K.; Arıkan, Orhan
The advances in convex optimization techniques have offered new formulations of design with improved control over the performance of FIR filters. By using lifting techniques, the design of a length-L FIR filter can be formulated as a convex semidefinite program (SDP) in terms of an L× L matrix that must be rank-1. Although this formulation provides means for introducing highly flexible design constraints on the magnitude and phase responses of the filter, convex solvers implementing interior point methods almost never provide a rank-1 solution matrix. To obtain a rank-1 solution, we propose a novel Directed Iterative Rank Refinement (DIRR) algorithm, where at each iteration a matrix is obtained by solving a convex optimization problem. The semidefinite cost function of that convex optimization problem favors a solution matrix whose dominant singular vector is on a direction determined in the previous iterations. Analytically it is shown that the DIRR iterations provide monotonic improvement, and the global optimum is a fixed point of the iterations. Over a set of design examples it is illustrated that the DIRR requires only a few iterations to converge to an approximately rank-1 solution matrix. The effectiveness of the proposed method and its flexibility are also demonstrated for the cases where in addition to the magnitude constraints, the constraints on the phase and group delay of filter are placed on the designed filter.
Open Access
Fir filter design by convex optimization using rank refinement
(2014) Dedeoğlu, Mehmet
Finite impulse response ﬁlters have been one of the primary topics of digital signal processing since their inception. Consequently, diverse class of design techniques including Chebyshev approximation, Fast Fourier Transform and optimization based methods had been proposed in the literature. With developments in com- putational tools, new design technique tools and formulations on ﬁlters including interior-point solvers and semideﬁnite programming (SDP), emerged. Since FIR ﬁlter design problem can be modelled as a quadratically constrained quadratic program, ﬁlter design problem can be solved via interior-point based convex op- timization methods such as semideﬁnite programming. Unfortunately, SDP for- mulation of problem is nonconvex due to positive lower limit constraint in the passband. To overcome that problem, nonconvex problem can be cast into a convex SDP using semideﬁnite relaxation, which can be solved in polynomial time. Since relaxed formulation does not guarantee rank-1 solution matrix, re- cently proposed directed iterative rank reﬁnement (DIRR) algorithm is used to impose a convex rank-1 constraint. Due to utilization of semideﬁnite relaxation and DIRR, addition of various constraints, such as phase and group delay masks, in convex manner is made possible. For feasibility type optimization formulations of ﬁlter design problem, a convergence rate improved version of DIRR is devel- oped. Proposed techniques are applied on ﬁlter design problems with diﬀerent set of constraints including phase and group delay constraints. Explicit simulations demostrate that the proposed technique is capable of solving nonlinear phase, phase constrained, and group delay constrained ﬁlter design problems.
Open Access
FIR filter design by iterative convex relaxations with rank refinement
(IEEE, 2014) Dedeoğlu, Mehmet; Alp, Yaşar Kemal; Arıkan, Orhan
Finite impulse response (FIR) filters have been a primary topic of digital signal processing since their inception. Although FIR filter design is an old problem, with the developments of fast convex solvers, convex modelling approach for FIR filter design has become an active research topic. In this work, we propose a new method based on convex programming for designing FIR filters with the desired frequency characteristics. FIR filter design problem, which is modelled as a non-convex quadratically constrained quadratic program (QCQP), is transformed to a semidefinite program (SDP). By relaxing the constraints, a convex programming problem, which we call RSDP(Relaxed Semidefinite Program), is obtained. Due to the relaxation, solution to the RSDPs fails to be rank-1. Typically used rank-1 approximations to the obtained RSDP solution does not satisfy the constraints. To overcome this issue, an iterative algorithm is proposed, which provides a sequence of solutions that converge to a rank-1 matrix. Conducted experiments and comparisons demonstrate that proposed method successfully designs FIR filters with highly flexible frequency characteristics.
Open Access
Interpolation for completely positive maps: Numerical solutions
(Societatea de Stiinte Matematice din Romania, 2018) Ambrozie, C.; Gheondea, Aurelian
We present a few techniques to find completely positive maps between full matrix algebras taking prescribed values on given data, based on semidefinite programming, convex minimization supported by a numerical example, as well as representations by linear functionals. The particular case of commutative data is also discussed.
Open Access
On extracting maximum stable sets in perfect graphs using Lovász's theta function
(Springer, 2006) Yıldırım, E. A.; Fan-Orzechowski, X.
We study the maximum stable set problem. For a given graph, we establish several transformations among feasible solutions of different formulations of Lovász's theta function. We propose reductions from feasible solutions corresponding to a graph to those corresponding to its induced subgraphs. We develop an efficient, polynomial-time algorithm to extract a maximum stable set in a perfect graph using the theta function. Our algorithm iteratively transforms an approximate solution of the semidefinite formulation of the theta function into an approximate solution of another formulation, which is then used to identify a vertex that belongs to a maximum stable set. The subgraph induced by that vertex and its neighbors is removed and the same procedure is repeated on successively smaller graphs. We establish that solving the theta problem up to an adaptively chosen, fairly rough accuracy suffices in order for the algorithm to work properly. Furthermore, our algorithm successfully employs a warm-start strategy to recompute the theta function on smaller subgraphs. Computational results demonstrate that our algorithm can efficiently extract maximum stable sets in comparable time it takes to solve the theta problem on the original graph to optimality.
Open Access
On robust solutions to linear least squares problems affected by data uncertainty and implementation errors with application to stochastic signal modeling
(Elsevier, 2004) Pınar, M. Ç.; Arıkan, Orhan
Engineering design problems, especially in signal and image processing, give rise to linear least squares problems arising from discretization of some inverse problem. The associated data are typically subject to error in these applications while the computed solution may only be implemented up to limited accuracy digits, i.e., quantized. In the present paper, we advocate the use of the robust counterpart approach of Ben-Tal and Nemirovski to address these issues simultaneously. Approximate robust counterpart problems are derived, which leads to semidefinite programming problems yielding stable solutions to overdetermined systems of linear equations affected by both data uncertainty and implementation errors, as evidenced by numerical examples from stochastic signal modeling.
Open Access
Optimal and robust power allocation for visible light positioning systems under illumination constraints
(IEEE, 2019-01) Keskin, Musa Furkan; Sezer, Ahmet Dündar; Gezici, Sinan
The problem of optimal power allocation among light emitting diode (LED) transmitters in a visible light positioning system is considered for the purpose of improving localization performance of visible light communication (VLC) receivers. Specifically, the aim is to minimize the Cramér-Rao lower bound (CRLB) on the localization error of a VLC receiver by optimizing LED transmission powers in the presence of practical constraints, such as individual and total power limitations and illuminance constraints. The formulated optimization problem is shown to be convex and thus can efficiently be solved via standard tools. We also investigate the case of imperfect knowledge of localization parameters and develop robust power allocation algorithms by taking into account both overall system uncertainty and individual parameter uncertainties related to the location and orientation of the VLC receiver. In addition, we address the total power minimization problem under predefined accuracy requirements to obtain the most energy-efficient power allocation vector for a given CRLB level. Numerical results illustrate the improvements in localization performance achieved by employing the proposed optimal and robust power allocation strategies over the conventional uniform and non-robust approaches.