Semidefinite Programming Research Articles

Security of discrete-modulated continuous-variable quantum key distribution

Continuous variable quantum key distribution with discrete modulation has the potential to provide information-theoretic security using widely available optical elements and existing telecom infrastructure. While their implementation is significantly simpler than that for protocols based on Gaussian modulation, proving their finite-size security against coherent attacks poses a challenge. In this work we prove finite-size security against coherent attacks for a discrete-modulated quantum key distribution protocol involving four coherent states and heterodyne detection. To do so, and contrary to most of the existing schemes, we first discretize all the continuous variables generated during the protocol. This allows us to use the entropy accumulation theorem, a tool that has previously been used in the setting of discrete variables, to construct the finite-size security proof. We then compute the corresponding finite-key rates through semi-definite programming and under a photon-number cutoff. Our analysis provides asymptotic rates in the range of 0.1−10−4 bits per round for distances up to hundred kilometres, while in the finite case and for realistic parameters, we get of the order of 10 Gbits of secret key after n∼1011 rounds and distances of few tens of kilometres.

Duality for non-smooth semidefinite multiobjective programming problems with equilibrium constraints using convexificators

Duality for non-smooth semidefinite multiobjective programming problems with equilibrium constraints using convexificators

On symmetry adapted bases in trigonometric optimization

The problem of computing the global minimum of a trigonometric polynomial is computationally hard. We address this problem for the case, where the polynomial is invariant under the exponential action of a finite group. The strategy is to follow an established relaxation strategy in order to obtain a converging hierarchy of lower bounds. Those bounds are obtained by numerically solving semi-definite programs (SDPs) on the cone of positive semi-definite Hermitian Toeplitz matrices, which is outlined in the book of Dumitrescu Dumitrescu (2007). To exploit the invariance, we show that the group has an induced action on the Toeplitz matrices and prove that the feasible region of the SDP can be restricted to the invariant matrices, whilst retaining the same solution. Then we construct a symmetry adapted basis tailored to this group action, which allows us to block-diagonalize invariant matrices and thus reduce the computational complexity to solve the SDP.The approach is in its generality novel for trigonometric optimization and complements the one that was proposed as a poster at the ISSAC 2022 conference Hubert et al. (2022) and later extended to Hubert et al. (2024). In the previous work, we first used the invariance of the trigonometric polynomial to obtain a classical polynomial optimization problem on the orbit space and subsequently relaxed the problem to an SDP. Now, we first make the relaxation and then exploit invariance.Partial results of this article have been presented as a poster at the ISSAC 2023 conference Metzlaff (2023).

Near-term quantum algorithm for solving the MaxCut problem with fewer quantum resources

MaxCut is an NP-hard combinatorial optimization problem in graph theory. The quantum approximate optimization algorithms (QAOAs) offer new methods for solving such problems, which are potentially better than classical schemes. However, the requirement of N qubits for solving graphs with N-vertex in QAOA, coupled with the large-N trainability issue due to barren plateaus, poses a substantial challenge for noisy intermediate-scale quantum (NISQ) devices. Recently, a hybrid quantum–classical algorithm has been proposed for solving semidefinite programs (SDPs), named NISQ SDP Solver (NSS). In this paper, we study the performance of the NSS for solving MaxCut problem via the introduction of a near-term quantum algorithm and execution of experimental simulations. Since the MaxCut problem admits a relaxation into an SDP formulation, the SDP can be solved using NSS, with a subsequent classical post-processing step converting the hybrid density matrix into a MaxCut solution. After performing these steps, the near-term quantum algorithm will also inherit the advantages of NSS. We analyze the algorithm as compared to QAOAs and find that the depth of the quantum circuit is independent of the number of edges on the graph. Our algorithm requires logN qubits and has O(1) circuit depth. This implies that it uses exponentially fewer qubits compared to QAOAs while also requiring a significantly reduced circuit depth to solve the MaxCut problem. To demonstrate the effectiveness of the NSS, we focused on 3-regular graphs, 9-regular graphs, and ER graphs. Numerical results indicate that the quantum algorithm achieves a comparable approximation ratio with classical methods by solving the original high dimensional problem within a lower dimensional ansatz space. Analysis under various initial states shows that the algorithm exhibits a remarkably stable performance.

A moment approach for entropy solutions of parameter-dependent hyperbolic conservation laws

We propose a numerical method to solve parameter-dependent scalar hyperbolic partial differential equations (PDEs) with a moment approach, based on a previous work from Marx et al. (2020). This approach relies on a very weak notion of solution of nonlinear equations, namely parametric entropy measure-valued (MV) solutions, satisfying linear equations in the space of Borel measures. The infinite-dimensional linear problem is approximated by a hierarchy of convex, finite-dimensional, semidefinite programming problems, called Lasserre’s hierarchy. This gives us a sequence of approximations of the moments of the occupation measure associated with the parametric entropy MV solution, which is proved to converge. In the end, several post-treatments can be performed from this approximate moments sequence. In particular, the graph of the solution can be reconstructed from an optimization of the Christoffel–Darboux kernel associated with the approximate measure, that is a powerful approximation tool able to capture a large class of irregular functions. Also, for uncertainty quantification problems, several quantities of interest can be estimated, sometimes directly such as the expectation of smooth functionals of the solutions. The performance of our approach is evaluated through numerical experiments on the inviscid Burgers equation with parametrised initial conditions or parametrised flux function.

Certifying Ground-State Properties of Many-Body Systems

A ubiquitous problem in quantum physics is to understand the ground-state properties of many-body systems. Confronted with the fact that exact diagonalization quickly becomes impossible when increasing the system size, variational approaches are typically employed as a scalable alternative: Energy is minimized over a subset of all possible states and then different physical quantities are computed over the solution state. Despite remarkable success, rigorously speaking, all that variational methods offer are upper bounds on the ground-state energy. On the other hand, so-called relaxations of the ground-state problem based on semidefinite programming represent a complementary approach, providing lower bounds to the ground-state energy. However, in their current implementation, neither variational nor relaxation methods offer provable bound on other observables in the ground state beyond the energy. In this work, we show that the combination of the two classes of approaches can be used to derive certifiable bounds on the value of any observable in the ground state, such as correlation functions of arbitrary order, structure factors, or order parameters. We illustrate the power of this approach in paradigmatic examples of 1D and 2D spin-1/2 Heisenberg models. To improve the scalability of the method, we exploit the symmetries and sparsity of the considered systems to reach sizes of hundreds of particles at much higher precision than previous works. Our analysis therefore shows how to obtain certifiable bounds on many-body ground-state properties beyond energy in a scalable way. Published by the American Physical Society 2024

Specific target pinning control of complex dynamical networks based on semidefinite programming strategy

Pinning control of complex networks aims to force the states of all nodes to reach the desired goal by controlling a few nodes in the network. The easiness of a pinning strategy can be measured by the minimum eigenvalue λ1 of the specific matrix L+Γ, where L is the Laplacian matrix of the underlying network and Γ is a diagonal matrix with Γi,i=1 if node i is pinned. A larger value λ1 indicates a smaller coupling strength. We here are interested in two key questions on this issue. The first one is, what is the minimum number of pinned nodes to achieve system synchronization when individual node dynamics and the coupling strength of the network are given? We show an upper bound of the minimum eigenvalue λ1, by which a lower bound of the number of pinned nodes is presented. The second one is, which nodes should be pinned such that the system achieves synchronization much easier if a potential pinning target node set S0 (instead of the whole node set) and the number of pinned nodes l (l<|S0| ) are both restricted beforehand? This specific target pinning control problem is raised for the first time. By transforming it into a semi-definite programming problem, we then use a convex optimization tool to solve this problem. Experimental results show that the strategy is more effective than other classical node selection indices.

Super-resolution delay-Doppler estimation for OTFS-based automotive radar

In this paper, we consider the problem of joint delay-Doppler estimation in an automotive radar based on orthogonal time frequency space (OTFS) modulation. We propose a super-resolution estimation approach that operates in the continuous domain and employs a two-dimensional atomic norm carrying delay and Doppler shift information to mitigate the off grid errors. The estimation task of target parameters is reformulated as a semidefinite program (SDP) problem. To solve this problem in an efficient way, we leverage the alternating direction method of multipliers (ADMM) and adopt an iterative procedure to obtain the optimal solution. The effectiveness of the proposed method is confirmed by theoretical analyses. Numerical results are presented to demonstrate the superior performance of the proposed method in comparison with the state-of-the-arts.

Higher-order Newton methods with polynomial work per iteration

We present generalizations of Newton's method that incorporate derivatives of an arbitrary order d but maintain a polynomial dependence on dimension in their cost per iteration. At each step, our dth-order method uses semidefinite programming to construct and minimize a sum of squares-convex approximation to the dth-order Taylor expansion of the function we wish to minimize. We prove that our dth-order method has local convergence of order d. This results in lower oracle complexity compared to the classical Newton method. We show on numerical examples that basins of attraction around local minima can get larger as d increases. Under additional assumptions, we present a modified algorithm, again with polynomial cost per iteration, which is globally convergent and has local convergence of order d.

Sketch-and-solve approaches to k-means clustering by semidefinite programming

Abstract We study a sketch-and-solve approach to speed up the Peng–Wei semidefinite relaxation of $k$-means clustering. When the data are appropriately separated we identify the $k$-means optimal clustering. Otherwise, our approach provides a high-confidence lower bound on the optimal $k$-means value. This lower bound is data-driven; it does not make any assumption on the data nor how they are generated. We provide code and an extensive set of numerical experiments where we use this approach to certify approximate optimality of clustering solutions obtained by k-means++.

Exact and Approximation Algorithms for Sparse Principal Component Analysis

Sparse principal component analysis (SPCA) is designed to enhance the interpretability of traditional principal component analysis by optimally selecting a subset of features that comprise the first principal component. Given the NP-hard nature of SPCA, most current approaches resort to approximate solutions, typically achieved through tractable semidefinite programs or heuristic methods. To solve SPCA to optimality, we propose two exact mixed-integer semidefinite programs (MISDPs) and an arbitrarily equivalent mixed-integer linear program. The MISDPs allow us to design an effective branch-and-cut algorithm with closed-form cuts that do not need to solve dual problems. For the proposed mixed-integer formulations, we further derive the theoretical optimality gaps of their continuous relaxations. Besides, we apply the greedy and local search algorithms to solving SPCA and derive their first-known approximation ratios. Our numerical experiments reveal that the exact methods we developed can efficiently find optimal solutions for data sets containing hundreds of features. Furthermore, our approximation algorithms demonstrate both scalability and near-optimal performance when benchmarked on larger data sets, specifically those with thousands of features. History: Accepted by Andrea Lodi, Area Editor for Design &amp; Analysis of Algorithms—Discrete. Funding: This research was supported in part by the Division of Civil, Mechanical and Manufacturing Innovation [Grant 224614], the Division of Computing and Communication Foundations [Grant 2246417], and the Office of Naval Research [Grant N00014-24-1-2066]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0372 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0372 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Code and Data Repository for Exact and Approximation Algorithms for Sparse Principal Component Analysis

Sparse principal component analysis (SPCA) is designed to enhance the interpretability of traditional principal component analysis by optimally selecting a subset of features that comprise the first principal component. Given the NP-hard nature of SPCA, most current approaches resort to approximate solutions, typically achieved through tractable semidefinite programs or heuristic methods. To solve SPCA to optimality, we propose two exact mixed-integer semidefinite programs (MISDPs) and an arbitrarily equivalent mixed-integer linear program. The MISDPs allow us to design an effective branch-and-cut algorithm with closed-form cuts that do not need to solve dual problems. For the proposed mixed-integer formulations, we further derive the theoretical optimality gaps of their continuous relaxations. Besides, we apply the greedy and local search algorithms to solving SPCA and derive their first-known approximation ratios. Our numerical experiments reveal that the exact methods we developed can efficiently find optimal solutions for data sets containing hundreds of features. Furthermore, our approximation algorithms demonstrate both scalability and near-optimal performance when benchmarked on larger data sets, specifically those with thousands of features.

Active Reconfigurable Intelligent Surface Enhanced Internet of Medical Things.

The incredible potentiality of reconfigurable intelligent surface (RIS) in addressing power supply and obstacle environment of Internet of Medical Things (IoMT) has been capturing our interest. Considering the nettlesome "double-fading" effect introduced by passive RIS, we investigate an active RIS-enhanced IoMT system in this article, where the wireless power transfer (WPT) from power station (PS) to IoMT devices and the wireless information transfer (WIT) from IoMT devices to the access point (AP) are both implemented with the assistance of active RIS. Aiming to maximize the sum throughput of the considered IoMT system, a joint design of time schedules and reflecting coefficient matrices of the active RIS is proposed. Trapped by the non-convex and obstinate optimization problem, we explore the semi-definite programming (SDP) relaxation and successive convex approximation (SCA) techniques based on alternating optimization (AO) algorithm. Simulation results verify our solution approach to the intractable optimization problem and showcase the boosted spectrum and energy efficiency of the active RIS-enhanced IoMT system.

Peak estimation of rational systems using convex optimization

This paper presents algorithms that upper-bound the peak value of a state function along trajectories of a continuous-time system with rational dynamics. The finite-dimensional but nonconvex peak estimation problem is cast as a convex infinite-dimensional linear program in occupation measures. This infinite-dimensional program is then truncated into finite-dimensions using the moment-Sum-of-Squares (SOS) hierarchy of semidefinite programs. Prior work on treating rational dynamics using the moment-SOS approach involves clearing dynamics to common denominators or adding lifting variables to handle reciprocal terms under new equality constraints. Our solution method uses a sum-of-rational method based on absolute continuity of measures. The Moment-SOS truncations of our program possess lower computational complexity and (empirically demonstrated) higher accuracy of upper bounds on example systems as compared to prior approaches.

Decentralized leader-following consensus control design for discrete-time multi-agent systems with switching topology

The problem of consensus control of linear discrete-time multi-agent systems (MASs) with switching topology is considered in the presence of a leader. The goal of consensus control is to bring the states of all agents to the leader state while providing stability for local agents, as well as the MAS as a whole. In contrast to the traditional approach, which uses the concept of an extended dynamic multi-agent system model and communication topology graph Laplacian, this paper proposes a decomposition approach, which provides a separate design of local controllers. The control law is chosen in the form of distributed feedback with discrete PID controllers. The problem of local controllers’ design is reduced to a set of semidefinite programming problems using the method of invariant ellipsoids. Sufficient conditions for agents’ stabilization and global consensus condition fulfillment are obtained using the linear matrix inequality technique. The availability of information about a finite set of possible configurations between agents allows us to design local controllers offline at the design stage. A numerical example demonstrates the effectiveness of the proposed approach.

Designing optimal protocols in Bayesian quantum parameter estimation with higher-order operations

Using quantum systems as sensors or probes has been shown to greatly improve the precision of parameter estimation by exploiting unique quantum features such as entanglement. A major task in quantum sensing is to design the optimal protocol, i.e., the most precise one. It has been solved for some specific instances of the problem, but in general even numerical methods are not known. Here, we focus on the single-shot Bayesian setting, where the goal is to find the optimal initial state of the probe (which can be entangled with an auxiliary system), the optimal measurement, and the optimal estimator function. We leverage the formalism of higher-order operations to develop a method based on semidefinite programming that finds a protocol that is close to the optimal one with arbitrary precision. Crucially, our method is not restricted to any specific quantum evolution, cost function, or prior distribution, and thus can be applied to any estimation problem. Moreover, it can be applied to both single-parameter or multiparameter estimation tasks. We demonstrate our method with three examples, consisting of unitary phase estimation, thermometry in a bosonic bath, and multiparameter estimation of an SU(2) transformation. Exploiting our methods, we extend several results from the literature. For example, in the thermometry case, we find the optimal protocol at any finite time and quantify the usefulness of entanglement. Published by the American Physical Society 2024

Convergence rate analysis of the gradient descent–ascent method for convex–concave saddle-point problems

In this paper, we study the gradient descent–ascent method for convex–concave saddle-point problems. We derive a new non-asymptotic global convergence rate in terms of distance to the solution set by using the semidefinite programming performance estimation method. The given convergence rate incorporates most parameters of the problem and it is exact for a large class of strongly convex-strongly concave saddle-point problems for one iteration. We also investigate the algorithm without strong convexity and we provide some necessary and sufficient conditions under which the gradient descent–ascent enjoys linear convergence.

Variational Quantum Algorithms for Semidefinite Programming

A semidefinite program (SDP) is a particular kind of convex optimization problem with applications in operations research, combinatorial optimization, quantum information science, and beyond. In this work, we propose variational quantum algorithms for approximately solving SDPs. For one class of SDPs, we provide a rigorous analysis of their convergence to approximate locally optimal solutions, under the assumption that they are weakly constrained (i.e., N&amp;#x226B;M, where N is the dimension of the input matrices and M is the number of constraints). We also provide algorithms for a more general class of SDPs that requires fewer assumptions. Finally, we numerically simulate our quantum algorithms for applications such as MaxCut, and the results of these simulations provide evidence that convergence still occurs in noisy settings.

Flexible dynamic boundary microgrid operation considering network and load unbalances

Flexible microgrids with dynamic boundaries have recently been introduced in the literature. With the ability to reconfigure the topology of the microgrids dynamically through remotely controlled switches, flexible microgrids with dynamic boundaries can further improve the resiliency and energy efficiency of microgrids with distributed energy resources (DERs). This paper focuses on the optimal operation considering one of the predominant characteristics of microgrids and distribution systems – unbalanced networks and loads. In existing literature, balanced modeling of microgrids is more common due to its attractive simplicity. The three-phase power unbalance has not been considered as a constraint on the generation units in a microgrid. Negative sequence constraints have also been neglected. In this article, we propose a set of constraints that is specifically related to the capabilities of inverter interfaced resources to supply unbalanced current/power when the microgrid is islanded from the main distribution grid. We incorporate the new set of constraints into two optimization formulations leveraging two convex relaxations of the three-phase power flow equations: mixed-integer linear programming (MILP) and mixed-integer semidefinite programming (MISDP) that optimize the dispatch of controllable switches and DERs in the microgrid. The algorithms are then extended to networked microgrids with grid-forming sources. We test the algorithms on a realistic community microgrid model in Puerto Rico as well as standardized IEEE distribution test feeders. The testing results demonstrate the performance of the proposed algorithms. The MILP is fast and scalable, and the MISDP enforces the negative sequence voltage constraints.

Resilient interval observer-based coordination control for multi-agent systems under stealthy attacks

In this paper, the coordination control problem of multi-agent systems (MASs) subjected to stealthy attacks and uncertainties is considered, where the uncertainties refer to unknown external disturbances and initial states. We consider a scenario in which the sensors are maliciously attacked, rendering the traditional interval observer’s bounds ineffective for estimating the system states. By solving the linear programming problem, the maximum destructive attack sequences are generated. A distributed resilient interval observer composed of a resilient framer and a control protocol is designed, in which the construction of control protocol for each agent depends on the framer information of its own and its neighbors. It is demonstrated by the cooperativity theory and the Lyapunov stability theory that the distributed resilient interval observer can not only realize interval-valued state estimation for each agent but also drive the cooperative behavior of MASs. In addition, the observer gain matrix is selected and optimized by solving a semi-definite programming problem to provide tighter bounds for each agent. Meanwhile, the bounds of stealthy attacks are calculated. Finally, through a simulation example, the superiority of the distributed resilient interval observer and the effectiveness of the resilient interval observer-based consensus control algorithms are validated.