Semidefinite Programming Research Articles

A powered descent trajectory planning method with quantitative consideration of safe distance to obstacle

High scientific value areas on celestial bodies such as the Moon and Mars are often located in hazardous terrains. To achieve safe landing exploration, a novel planning method is proposed, which can ensure that the planned trajectory maintains a user-specified distance from obstacles, thus reducing potential collision risk induced by factors such as the body size and model uncertainties. Firstly, the basic model of the trajectory optimization problem and its convexification version is given. The obstacles are modeled as polynomial functions, based on which the “if-then” obstacle avoidance logic explicitly considering the safe distance, is described as a sum of squares constraint. This constraint formulation applies to any obstacle described by a finite number of polynomials, independent of the specific expression of the polynomials (reflecting the shape of the obstacle). Subsequently, the convexification process for the obstacle avoidance constraint is given. Finally, the sequential sum of squares programming problem for the obstacle avoidance trajectory is established, which boils down to a series of semidefinite programming problems. Simulation results show that the closest distance between the planned trajectory and obstacles strictly satisfies the specified distance constraint, and the trajectory could avoid non-convex obstacles. With the promising convergence properties of underlying convex optimization algorithms, advanced autonomous obstacle avoidance guidance schemes are expected to be formed based on the proposed trajectory planning method.

On parametric semidefinite programming with unknown boundaries

In this paper, we study parametric semidefinite programs (SDPs) where the solution space of both the primal and dual problems change simultaneously. Given a bounded set, we aim to find the a priori unknown maximal permissible perturbation set within it where the semidefinite program problem has a unique optimum and is analytic with respect to the parameters. Our approach reformulates the parametric SDP as a system of partial differential equations (PDEs) where this maximal analytical permissible set (MAPS) is the set on which the system of PDEs is well-posed. A sweeping Euler scheme is developed to approximate this a priori unknown perturbation set. We prove local and global error bounds for this second-order sweeping Euler scheme and demonstrate the method in comparison to existing SDP solvers and its performance on several two-parameter and three-parameter SDPs for which the MAPS can be visualized.

Efficient symbol level precoding design for full-duplex integrated sensing and communication systems

Full-duplex (FD) integrated sensing and communication (ISAC) system attracts much attention because adopting the FD mode can enhance the transmission efficiency and reduce the transmission time of the ISAC system. However, the inevitable self-interference (SI) in FD mode would result in severe performance degradation. This paper considers the FD ISAC system and investigates a symbol-level precoding (SLP) design, which minimizes the Cramér-Rao Bound of target angle estimation while considering the constant-envelope power restraint, SI suppression and the quality of service requirement. Due to the non-convex fractional problem and the unit-modulus constraint, the optimization problem is reformulated by implementing the Schur complement, where the optimal waveform is obtained via the semidefinite programming algorithm. The SLP design that optimizes each symbol vector for all users may suffer high computational complexity of matrix multiplication with high numbers of users. A grouped SLP strategy in which the communication users are divided into several groups on the basis of channel correlation is proposed to reduce the complexity. The proposed grouped SLP design only suppresses group-to-group interference and converts the intra-group interference into a constructive interference, which can reduce the dimension of each precoded signal significantly, resulting in a low computational complexity. Simulation results illustrate that the proposed SLP design can effectively mitigate SI, which can improve the accuracy of target detection. Moreover, the proposed grouped-SLP design can further reduce computational complexity without dramatic performance degradation.

Multi-antenna dual-blind deconvolution for joint radar-communications via SoMAN minimization

In joint radar-communications (JRC) applications such as secure military receivers, often the radar and communications signals are overlaid in the received signal. In these passive listening outposts, the signals and channels of both radar and communications are unknown to the receiver. The ill-posed problem of recovering all signal and channel parameters from the overlaid signal is termed as dual-blind deconvolution (DBD). In this work, we investigate DBD for a multi-antenna receiver. We model the radar and communications channels with a few (sparse) continuous-valued parameters such as time delays, Doppler velocities, and directions-of-arrival (DoAs). To solve this highly ill-posed DBD, we propose to minimize the sum of multivariate atomic norms (SoMAN) that depend on unknown parameters. To this end, we devise an exact semidefinite program using theories of positive hyperoctant trigonometric polynomials (PhTP). Our theoretical analyses show that the minimum number of samples and antennas required for perfect recovery is logarithmically dependent on the maximum of the number of radar targets and communications paths rather than their sum. We show that our approach is easily generalized to include several practical issues such as gain/phase errors and additive noise. Numerical experiments show the exact parameter recovery for different JRC scenarios.

Characterizations of ε-Approximate Solutions for Robust Convex Semidefinite Programming Problems

Characterizations of ε-Approximate Solutions for Robust Convex Semidefinite Programming Problems

Enhanced angle estimation in MIMO radar: Combine RD-MUSIC and SDP optimization

Multiple Input Multiple Output (MIMO) radars are increasingly employed. However, with a growing array of antenna components, the dimension of the sampling covariance matrix and the computational complexity of traditional Direction of Arrival (DOA) and Direction of Departure (DOD) estimation algorithms increase exponentially. In order to accurately and effectively identify and locate signal sources and to optimize the computational complexity, this paper presents an innovative approach for detecting target numbers based on a Semi-Definite Programming (SDP) problem, which generates efficient solutions even for higher dimensions. In addition, the SDP’s robustness is developed through shrinkage adaptation using Large Covariance Matrices (LCMs), which are crucial for angle estimation. Following the effective detection of a target, the tapering estimation is applied to the LCMs under the computationally efficient Reduced-Dimension Multiple Signal Classification (RD-MUSIC) algorithm, which is more accurate and less computationally intensive than 2D search algorithms. Simulation outcomes demonstrate that the efficacy of our proposed model achieves high success rates in target detection and enhanced precision in DOA and DOD estimations.

Robust Beamforming for Downlink Multi-Cell Systems: A Bilevel Optimization Perspective

Utilization of inter-base station cooperation for information processing has shown great potential in enhancing the overall quality of communication services (QoS) in wireless communication networks. Nevertheless, such cooperations require the knowledge of channel state information (CSI) at base stations (BSs), which is assumed to be perfectly known. However, CSI errors are inevitable in practice which necessitates beamforming technique that can achieve robust performance in the presence of channel estimation errors. Existing approaches relax the robust beamforming design problems into semidefinite programming (SDP), which can only achieve a solution that is far from being optimal. To this end, this paper views robust beamforming design problems from a bilevel optimization perspective. In particular, we focus on maximizing the worst-case weighted sum-rate (WSR) in the downlink multi-cell multi-user multiple-input single-output (MISO) system considering bounded CSI errors. We first reformulate this problem into a bilevel optimization problem and then develop an efficient algorithm based on the cutting plane method. A distributed optimization algorithm has also been developed to facilitate the parallel processing in practical settings. Numerical results are provided to confirm the effectiveness of the proposed algorithm in terms of performance and complexity, particularly in the presence of CSI uncertainties.

Accelerated first-order methods for a class of semidefinite programs

Accelerated first-order methods for a class of semidefinite programs

N-representable one-electron reduced density matrix reconstruction with frozen core electrons.

Recent advances in quantum crystallography have shown that, beyond conventional charge density refinement, a one-electron reduced density matrix (1-RDM) satisfying N-representability conditions can be reconstructed using jointly experimental X-ray structure factors and directional Compton profiles (DCP) through semidefinite programming. So far, such reconstruction methods for 1-RDM, not constrained to idempotency, have been tested only on a toy model system (CO2). In this work, a new method is assessed on crystalline urea [CO(NH2)2] using static (0 K) and dynamic (50 K) artificial experimental data. An improved model, including symmetry constraints and frozen core-electron contribution, is introduced to better handle the increasing system complexity. Reconstructed 1-RDMs, deformation densities and DCP anisotropy are analysed, and it is demonstrated that the changes in the model significantly improve the reconstruction quality, even when there is insufficient information and data corruption. The robustness of the model and the strategy are thus shown to be well adapted to address the reconstruction problem from actual experimental scattering data.

Distributionally robust chance-constrained optimization for the integrated berth allocation and quay crane assignment problem

In this paper, we are the first to propose a distributionally robust chance-constrained (DRCC) optimization model for the integrated berth allocation and quay crane assignment problem (BACAP) in container terminals. In contrast to the classical deterministic BACAP model, we consider the arrival time to be uncertain due to the frequent arrival delays in ports. We then impose a chance constraint that the service time must start after the uncertain arrival time with a probability of at least 1−ϵ, where 1−ϵ represents the target service level in container terminals (ϵ represents the target risk tolerance of the port manager). Under the moment-based ambiguity set, we reformulate the DRCC model into a mixed integer semi-definite programming (MISDP) model. Additionally, we develop an efficient decomposition branch-and-bound algorithm to solve the MISDP model and obtain the exact solution. Fortunately, a special case of the DRCC model arises when the mean and covariance utilized in the ambiguity set are precise, allowing for the transformation of the DRCC model into a mixed integer programming (MIP) model. This conversion significantly reduces the complexity of the problem. Impressively, the solving time of the MISDP model with the decomposition branch-and-bound algorithm is comparable to that of the transformed MIP model. The numerical results show that our model can achieve a schedule with high service at low cost. Meanwhile, we have made an intriguing discovery that the correlation between the target risk tolerance and the actual service level can be depicted as a staircase function regardless of the datasets. This finding offers crucial insights for port management, enabling them to strike a balance between the cost and actual service level by determining an appropriate target service level.

On Integrality in Semidefinite Programming for Discrete Optimization

On Integrality in Semidefinite Programming for Discrete Optimization

Terminal Embeddings in Sublinear Time

Recently (Elkin, Filtser, Neiman 2017) introduced the concept of a {\it terminal embedding} from one metric space $(X,d_X)$ to another $(Y,d_Y)$ with a set of designated terminals $T\subset X$. Such an embedding $f$ is said to have distortion $\rho\ge 1$ if $\rho$ is the smallest value such that there exists a constant $C>0$ satisfying \begin{equation*} \forall x\in T\ \forall q\in X,\ C d_X(x, q) \le d_Y(f(x), f(q)) \le C \rho d_X(x, q) . \end{equation*} When $X,Y$ are both Euclidean metrics with $Y$ being $m$-dimensional, recently (Narayanan, Nelson 2019), following work of (Mahabadi, Makarychev, Makarychev, Razenshteyn 2018), showed that distortion $1+\epsilon$ is achievable via such a terminal embedding with $m = O(\epsilon^{-2}\log n)$ for $n := |T|$. This generalizes the Johnson-Lindenstrauss lemma, which only preserves distances within $T$ and not to $T$ from the rest of space. The downside of prior work is that evaluating their embedding on some $q\in \mathbb{R}^d$ required solving a semidefinite program with $\Theta(n)$ constraints in~$m$ variables and thus required some superlinear $\mathrm{poly}(n)$ runtime. Our main contribution in this work is to give a new data structure for computing terminal embeddings. We show how to pre-process $T$ to obtain an almost linear-space data structure that supports computing the terminal embedding image of any $q\in\mathbb{R}^d$ in sublinear time $O^* (n^{1-\Theta(\epsilon^2)} + d)$. To accomplish this, we leverage tools developed in the context of approximate nearest neighbor search.

Semidefinite Approximations for Bicliques and Bi-Independent Pairs

We investigate some graph parameters dealing with bi-independent pairs (A, B) in a bipartite graph [Formula: see text], that is, pairs (A, B) where [Formula: see text], and [Formula: see text] are independent. These parameters also allow us to study bicliques in general graphs. When maximizing the cardinality [Formula: see text], one finds the stability number [Formula: see text], well-known to be polynomial-time computable. When maximizing the product [Formula: see text], one finds the parameter g(G), shown to be NP-hard by Peeters in 2003, and when maximizing the ratio [Formula: see text], one finds h(G), introduced by Vallentin in 2020 for bounding product-free sets in finite groups. We show that h(G) is an NP-hard parameter and, as a crucial ingredient, that it is NP-complete to decide whether a bipartite graph G has a balanced maximum independent set. These hardness results motivate introducing semidefinite programming (SDP) bounds for g(G), h(G), and [Formula: see text] (the maximum cardinality of a balanced independent set). We show that these bounds can be seen as natural variations of the Lovász ϑ-number, a well-known semidefinite bound on [Formula: see text]. In addition, we formulate closed-form eigenvalue bounds, and we show relationships among them as well as with earlier spectral parameters by Hoffman and Haemers in 2001 and Vallentin in 2020. Funding: This work was supported by H2020 Marie Skłodowska-Curie Actions [Grant 813211 (POEMA)].

The Chvátal–Gomory procedure for integer SDPs with applications in combinatorial optimization

AbstractIn this paper we study the well-known Chvátal–Gomory (CG) procedure for the class of integer semidefinite programs (ISDPs). We prove several results regarding the hierarchy of relaxations obtained by iterating this procedure. We also study different formulations of the elementary closure of spectrahedra. A polyhedral description of the elementary closure for a specific type of spectrahedra is derived by exploiting total dual integrality for SDPs. Moreover, we show how to exploit (strengthened) CG cuts in a branch-and-cut framework for ISDPs. Different from existing algorithms in the literature, the separation routine in our approach exploits both the semidefinite and the integrality constraints. We provide separation routines for several common classes of binary SDPs resulting from combinatorial optimization problems. In the second part of the paper we present a comprehensive application of our approach to the quadratic traveling salesman problem (QTSP). Based on the algebraic connectivity of the directed Hamiltonian cycle, two ISDPs that model the QTSP are introduced. We show that the CG cuts resulting from these formulations contain several well-known families of cutting planes. Numerical results illustrate the practical strength of the CG cuts in our branch-and-cut algorithm, which outperforms alternative ISDP solvers and is able to solve large QTSP instances to optimality.

Network partition and distributed voltage coordination control strategy of active distribution network system considering photovoltaic uncertainty

Aiming at the voltage exceeding problem caused by the introduction of Photovoltaic (PV) in Active distribution network (ADN), a comprehensive electrical distance index was established considering active and reactive voltage sensitivity, and the uncertainty of traditional PV probability density function (PDF) was optimized by random simulation method. The electrical distance expectation matrix between nodes was established by discretization of PV output probability distribution characteristics, and the ADN system was partitioned by affinity propagation (AP) algorithm. Taking active power loss as the objective function, the non-convex reactive power optimization control problem is transformed into a convex quadratic programming problem by LinDistFlow equation. An improved alternating direction method of multipliers (ADMM) which adaptively adjusts the penalty parameters is used to convert the finite boundaries of adjacent regions into data. By coordinating on-load tap-changer (OLTC), capacitor banks (CBs), and PV inverters on different time scales, the fast optimal control of global voltage in ADN is realized. The proposed method is tested on IEEE 33-bus and IEEE 123-bus distribution systems. After the optimization control algorithm described in this paper, the system network loss was reduced by 3.88%. Compared with the semi-definite programming relaxation, the calculation speed of LinDistFlow equation is increased by 72.2%. The results fully verify the feasibility and high efficiency of the control strategy described in this paper.

How Do Exponential Size Solutions Arise in Semidefinite Programming?

How Do Exponential Size Solutions Arise in Semidefinite Programming?

Radius of information for two intersected centered hyperellipsoids and implications in optimal recovery from inaccurate data

For objects belonging to a known model set and observed through a prescribed linear process, we aim at determining methods to recover linear quantities of these objects that are optimal from a worst-case perspective. Working in a Hilbert setting, we show that, if the model set is the intersection of two hyperellipsoids centered at the origin, then there is an optimal recovery method which is linear. It is specifically given by a constrained regularization procedure whose parameters can be precomputed by semidefinite programming. This general framework can be applied to several scenarios, including the two-space problem and problems involving ℓ2-inaccurate data. It can also be applied to the problem of recovery from ℓ1-inaccurate data. For the latter, we reach the conclusion of existence of an optimal recovery method which is linear, again given by constrained regularization, under a computationally verifiable sufficient condition.

Parametric Semidefinite Programming: Geometry of the Trajectory of Solutions

In many applications, solutions of convex optimization problems are updated on-line, as functions of time. In this paper, we consider parametric semidefinite programs, which are linear optimization problems in the semidefinite cone whose coefficients (input data) depend on a time parameter. We are interested in the geometry of the solution (output data) trajectory, defined as the set of solutions depending on the parameter. We propose an exhaustive description of the geometry of the solution trajectory. As our main result, we show that only six distinct behaviors can be observed at a neighborhood of a given point along the solution trajectory. Each possible behavior is then illustrated by an example. Funding: This work was supported by OP RDE [Grant CZ.02.1.01/0.0/0.0/16_019/0000765].

Detecting entanglement by pure bosonic extension

In the realm of quantum information theory, the detection and quantification of quantum entanglement stand as paramount tasks. The relative entropy of entanglement (REE) serves as a prominent measure of entanglement, with extensive applications spanning numerous related fields. The positive partial transpose (PPT) criterion, while providing an efficient method for the computation of REE, unfortunately, falls short when dealing with bound entanglement. In this study, we propose a method termed “pure bosonic extension” to enhance the practicability of k-bosonic extensions, which approximates the set of separable states from the “outside”, through a hierarchical structure. It enables efficient characterization of the set of k-bosonic extendible states, facilitating the derivation of accurate lower bounds for REE. Compared to the semi-definite programming (SDP) approach, such as the symmetric/bosonic extension function in QETLAB, our algorithm supports much larger dimensions and higher values of extension k. Published by the American Physical Society 2024

On semidefinite descriptions for convex hulls of quadratic programs

Quadratically constrained quadratic programs (QCQPs) are a highly expressive class of nonconvex optimization problems. While QCQPs are NP-hard in general, they admit a natural convex relaxation via the standard semidefinite program (SDP) relaxation. In this paper we study when the convex hull of the epigraph of a QCQP coincides with the projected epigraph of the SDP relaxation. We present a sufficient condition for convex hull exactness and show that this condition is further necessary under an additional geometric assumption. The sufficient condition is based on geometric properties of Γ, the cone of convex Lagrange multipliers, and its relatives Γ1 and Γ∘.