Positive Semidefinite Relaxation Research Articles

Sum-Rate Maximization Scheme for Multi-RIS-Assisted NOMA Uplink Systems

Sum-Rate Maximization Scheme for Multi-RIS-Assisted NOMA Uplink Systems

Prepositioning network design for humanitarian relief purposes under correlated demand uncertainty

Prepositioning relief network is an effective strategy to mitigate the impact of natural disasters and public health emergencies, such as the COVID-19 pandemic. However, designing a proper network is challenging due to limited information and, more importantly, the correlated demand uncertainty that exists among affected areas. We consider a network design problem for humanitarian relief purposes, where demand correlations exist and demand information is limited, i.e., only the mean demand and covariance matrix are known. Note that the covariance matrix can explicitly capture the correlated demand among areas. We formulate this problem as a mixed-integer two-stage distributionally robust location-inventory model, which is generally NP-hard and computationally intractable. The model is further reformulated as a mixed-integer conic problem based on copositive cones, and it is tractable with positive semidefinite relaxation. To accelerate the problem-solving process, we design an interpretable branching-and-pricing heuristic with a warm start. Both semi-case study and simulation results demonstrate that explicitly modelling demand correlation can decrease unmet demand.

Secure wireless communications in the multi-user MISO interference channel assisted by multiple reconfigurable intelligent surfaces

This paper exploits reconfigurable intelligent surfaces (RIS) to enhance physical layer security (PLS) in a challenging radio environment. By adjusting the reflecting coefficients, RIS can provide a programmable wireless environment so that the electromagnetic wave can propagate in the desired way. Specifically, we consider a scenario where multiple user pairs communicate simultaneously over the same channel in a multiuser MISO interference channel and each pair of transceivers keeps their confidential information secret from other receivers. We investigate the secrecy rate balance of the system aided by multiple RISs and exploit the additional design degrees of freedom provided by the coordinated RISs to increase the secrecy capacity, achieving secrecy transmission. In particular, we adopt cooperative beamforming among the collaborative transmitters to maximize all users' minimum secrecy rate, achieving the secrecy rate balance. Due to the non-convexity of the formulated optimization problem, we propose alternating optimization (AO) to solve it iteratively and optimize active the transmit beamforming at the transmitters and passive phase shifts at the RISs based on positive semi-definite relaxation (SDR) and successive convex approximation (SCA). In each iteration, we solve a semi-definite programming (SDP) problem and finally get the optimal local solution to the optimization problem. Finally, the simulation results validate that the proposed algorithm is effective and the introduced RISs can significantly enhance the system secrecy performance compared to conventional baseline schemes.

Radar Target Detection Based on Waveform Design under Atmospheric Attenuation

Under the condition of atmospheric attenuation, the mismatch of radar echo signals leads to the decrease of signal-to-noise ratio (SNR) and detection probability. A joint optimization algorithm for transmitting signal and receiving filter of MIMO radar with atmospheric system response is presented in this paper. Considering the energy constraint, similarity constraint and spectral coexistence constraint of the transmitting signal, and introducing the atmospheric channel response matrix, the echo model under the atmospheric response condition is constructed. By optimizing the transmitting signal and the receiving filter, the echo signal SNR and radar detection probability can be improved. On this basis, the ARMA stochastic model is used to simulate the atmospheric system response. Solve non-convex problems by matrix optimization, positive semi-definite relaxation, Charnes-Cooper transformation, and loop optimization methods. Finally, the rank-one matrix decomposition method is used to extract the optimal signal from the signal autocorrelation matrix. And we analyze the convergence of the algorithm, the degree of SNR improvement and the detection probability. Through numerical experiments, it is proved that after the atmospheric channel response is introduced and matched correctly, the SNR of the echo signal can be improved and the radar detection probability can be greatly increased.

A completely positive formulation of the graph isomorphism problem and its positive semidefinite relaxation

Aurora and Mehta (J Comb Optim 36(3):965–1006, 2018) show that two graphs $$G_1,G_2$$ , on n vertices each, are isomorphic if and only if the feasible region of a certain linear program, LP-GI, intersects with the Quadratic Assignment Problem (QAP)-polytope in $${\mathbb {R}}^{(n^4+n^2)/2}$$ . The linear program LP-GI in Aurora and Mehta (J Comb Optim 36(3):965–1006, 2018) is obtained by relaxing an integer linear program whose feasible points correspond to the isomorphisms between $$G_1,G_2$$ . In this paper we take an analogous approach with the linear programs replaced with conic programs. A completely positive description of the QAP-polytope was obtained in Povh and Rendl (Discrete Optim 6(3):231–241, 2009). By adding the graph conditions to this description we get a completely positive formulation of the graph isomorphism problem. However, analogous to integer linear programs, it is NP-hard to optimize over the cone of completely positive matrices. So we relax this formulation by replacing the cone of completely positive matrices with the cone of positive semidefinite matrices. We observe that the resulting SDP is the Lovasz Theta function (Lovasz in IEEE Trans Inf Theory 25(1):1–7, 1979) of a graph product of $$G_1,G_2$$ and can be efficiently computed. We provide a natural heuristic that uses the SDP to solve the graph isomorphism problem. We run our heuristic on several pairs of non-isomorphic strongly regular graphs and find the results to be encouraging. Further, by adding the non-negativity constraints to the SDP, we obtain a doubly non-negative formulation, DNN-GI. We show that if the set of optimal points in DNN-GI contains a point of rank at most 3, then the given pair of graphs must be isomorphic.

Completely positive and completely positive semidefinite tensor relaxations for polynomial optimization

Completely positive (CP) tensors, which correspond to a generalization of CP matrices, allow to reformulate or approximate a general polynomial optimization problem (POP) with a conic optimization problem over the cone of CP tensors. Similarly, completely positive semidefinite (CPSD) tensors, which correspond to a generalization of positive semidefinite (PSD) matrices, can be used to approximate general POPs with a conic optimization problem over the cone of CPSD tensors. In this paper, we study CP and CPSD tensor relaxations for general POPs and compare them with the bounds obtained via a Lagrangian relaxation of the POPs. This shows that existing results in this direction for quadratic POPs extend to general POPs. Also, we provide some tractable approximation strategies for CP and CPSD tensor relaxations. These approximation strategies show that, with a similar computational effort, bounds obtained from them for general POPs can be tighter than bounds for these problems obtained by reformulating the POP as a quadratic POP, which subsequently can be approximated using CP and PSD matrices. To illustrate our results, we numerically compare the bounds obtained from these relaxation approaches on small scale fourth-order degree POPs.

Yield-aware mask assignment by positive semidefinite relaxation in triple patterning using cut process

LELECUT type triple patterning lithography is one of the most promising techniques in 14 nm logic node and beyond. To prevent yield loss caused by overlay error, LELECUT mask assignment, which is tolerant to overlay error, is desired. We propose a method that obtains a LELECUT assignment that is tolerant to overlay error. The proposed method uses positive semidefinite relaxation and randomized rounding technique. In our method, the cost function that takes the length of boundary of features determined by the cut mask into account is introduced.

On linear conic relaxation of discrete quadratic programs

A special reformulation-linearization technique based linear conic relaxation is proposed for discrete quadratic programming (DQP). We show that the proposed relaxation is tighter than the traditional positive semidefinite programming relaxation. More importantly, when the proposed relaxation problem has an optimal solution with rank one or two, optimal solutions to the original DQP problem can be explicitly generated. This rank-two property is further extended to binary quadratic optimization problems and linearly constrained DQP problems. Numerical results indicate that the proposed relaxation is capable of providing high quality and robust lower bounds for DQP.

A new approximate algorithm for the Chebyshev center

The parameter or state estimation with bounded noises is getting increasingly important in many applications of practical systems with some uncertainties. The problem to estimate a deterministic parameter or state which is known to lie in an intersection of some ellipsoids can be formulated to find the Chebyshev center of the intersection set in the case of l2 norm of the estimation error. In this paper, an appropriate positive semidefinite relaxation of non-convex optimization problem is derived, and then a new algorithm for robust minimax estimation is provided. Some examples are given to compare the approximate estimate with the existing relaxed Chebyshev center.

Positive semidefinite relaxations for distance geometry problems

We consider distance geometry problems for determiningr-dimensional coordinates ofn points from a given set of the distances between the points. This problem is a fundamental problem in molecular biology for finding the structure of proteins from NMR (Nuclear Magnetic Resonance) data.

Solving Large-Scale Sparse Semidefinite Programs for Combinatorial Optimization

We present a dual-scaling interior-point algorithm and show how it exploits the structure and sparsity of some large-scale problems. We solve the positive semidefinite relaxation of combinatorial and quadratic optimization problems subject to boolean constraints. We report the first computational results of interior-point algorithms for approximating maximum cut semidefinite programs with dimension up to 3,000.

On a positive semidefinite relaxation of the cut polytope

We study the convex set L n defined by L n Z≔ { X| X = ( x ij ) a positive semidefinite n × n matrix, x ii = 1 for all i}. We describe several geometric properties of L n . In particular, we show that L n has 2 n − 1 vertices, which are its rank one matrices, corresponding to all bipartitions of the set {1, 2, …, n}. Our main motivation for investigating the convex set L n comes from combinatorial optimization, namely from approximating the max-cut problem. An important property of L n is that, due to the positive semidefinite constraints, one can optimize over it in polynomial time. On the other hand, L n still inherits the difficult structure of the underlying combinatorial problem. In particular, it is NP-hard to decide whether the optimum of the problem min Tr( CX), X ∈ L n is reached at a vertex. This result follows from the complete characterization of the matrices C of the form C = bb t for some vector b, for which the optimum of the above program is reached at a vertex.