Solution Of Semidefinite Program Research Articles

Power Grid AC-Based State Estimation: Vulnerability Analysis Against Cyber Attacks

To ensure grid efficiency and reliability, power system operators continuously monitor the operational characteristics of the grid through a critical process called state estimation (SE), which performs the task by filtering and fusing various measurements collected from grid sensors. This study analyzes the vulnerability of the key operation module, namely ac-based SE, against potential cyber attacks on data integrity, also known as false data injection attack (FDIA). A general form of FDIA can be formulated as an optimization problem, whose objective is to find a stealthy and sparse data injection vector on the sensor measurements with the aim of making the state estimate spurious and misleading. Due to the nonlinear ac measurement model and the cardinality constraint, the problem includes both continuous and discrete nonlinearities. To solve the FDIA problem efficiently, we propose a novel convexification framework based on semidefinite programming (SDP). By analyzing a globally optimal SDP solution, we delineate the “attackable region” for any given set of measurement types and grid topology, where the spurious state can be falsified by FDIA. Furthermore, we prove that the attack is stealthy and sparse, and derive performance bounds. Simulation results on various IEEE test cases indicate the efficacy of the proposed convexification approach. From the grid protection point of view, the results of this study can be used to design a security metric for the current practice against cyber attacks, redesign the bad data detection scheme, and inform proposals of grid hardening. From a theoretical point of view, the proposed framework can be used for other nonconvex problems in power systems and beyond.

Semidefinite programming solution of economic dispatch problem with non-smooth, non-convex cost functions

The paper presents a solution to economic dispatch (ED) problems with non-convex, non-smooth fuel cost functions, which characterize practical generating units. A method involving a unified semidefinite programming (SDP) formulation of different ED problems through cost function decomposition was presented. The solution of the resulting rank-relaxed SDP problem was refined to achieve the rank constraint using the method of convex iteration and branch-and-bound technique. The SDP method was investigated on some test problems in the literature. The results showed that the SDP method compared favorably with other methods, and can efficiently solve non-convex and non-smooth ED problems.

Moment-SOS relaxation of the medium term hydrothermal dispatch problem

This paper analyzes the application of moment-SOS (sum of squares) relaxation to solve the medium term hydrothermal dispatch (MTHD) problem for predominantly hydro systems. The optimal dispatch is formulated as a two stage stochastic programming problem, which is nonconvex due to constraints that represent hydro power production, and may have multiple solutions. The size and conditioning of the problem pose challenges to the use of the semidefinite programming (SDP) techniques that are discussed in the text. The size of the problem restricts this study to first order moment relaxations. Therefore, to improve the quality of relaxed solutions, additional meaningful constraints are introduced in the SDP problems. Subsequently, the problem re-scalded to achieve numerical stability in SDP relaxation. Results obtained for different test systems, most of them being partitions of the Brazilian hydroelectric system, indicate that the procedures used to improve the quality of SDP solutions are effective.

Bounding Averages Rigorously Using Semidefinite Programming: Mean Moments of the Lorenz System

We describe methods for proving bounds on infinite-time averages in differential dynamical systems. The methods rely on the construction of nonnegative polynomials with certain properties, similarly to the way nonlinear stability can be proved using Lyapunov functions. Nonnegativity is enforced by requiring the polynomials to be sums of squares, a condition which is then formulated as a semidefinite program (SDP) that can be solved computationally. Although such computations are subject to numerical error, we demonstrate two ways to obtain rigorous results: using interval arithmetic to control the error of an approximate SDP solution, and finding exact analytical solutions to relatively small SDPs. Previous formulations are extended to allow for bounds depending analytically on parametric variables. These methods are illustrated using the Lorenz equations, a system with three state variables $(x,y,z)$ and three parameters $(\beta,\sigma,r)$. Bounds are reported for infinite-time averages of all eighteen moments $x^ly^mz^n$ up to quartic degree that are symmetric under $(x,y)\mapsto(-x,-y)$. These bounds apply to all solutions regardless of stability, including chaotic trajectories, periodic orbits, and equilibrium points. The analytical approach yields two novel bounds that are sharp: the mean of $z^3$ can be no larger than its value of $(r-1)^3$ at the nonzero equilibria, and the mean of $xy^3$ must be nonnegative. The interval arithmetic approach is applied at the standard chaotic parameters to bound eleven average moments that all appear to be maximized on the shortest periodic orbit. Our best upper bound on each such average exceeds its value on the maximizing orbit by less than 1%. Many bounds reported here are much tighter than would be possible without computer assistance.

Semidefinite Programming Solution to the Spacecraft Analysis and Control Problem

This paper provides a semidefinite optimization solution to the analysis and control of a rigid rotating spacecraft. Due to the quaternion representation of a rotating body, the nonlinearities that describe the system are polynomial in nature and may be expressed in the Differential Algebraic Representation (DAR). Once the system is in this form, it is possible to apply linear-based tools for the analysis and control design of the closed loop dynamics. The task is then reduced to that of a semidefinite optimization problem subject to constraints in the form of Linear Matrix Inequalities (LMIs).

Convex Relaxation for Optimal Distributed Control Problems

This paper is concerned with the optimal distributed control (ODC) problem for linear discrete-time deterministic and stochastic systems. The objective is to design a static distributed controller with a prespecified structure that is globally optimal with respect to a quadratic cost functional. It is shown that this NP-hard problem has a quadratic formulation, which can be relaxed to a semidefinite program (SDP). If the SDP relaxation has a rank-1 solution, a globally optimal distributed controller can be recovered from this solution. By utilizing the notion of treewidth, it is proved that the nonlinearity of the ODC problem appears in such a sparse way that an SDP relaxation of this problem has a matrix solution with rank at most 3. Since the proposed SDP relaxation is computationally expensive for a large-scale system, a computationally cheap SDP relaxation is also developed with the property that its objective function indirectly penalizes the rank of the SDP solution. Various techniques are proposed to approximate a low-rank SDP solution with a rank-1 matrix, leading to near globally optimal controllers together with a bound on the optimality degree of each controller. The above results are developed for both finite-horizon and infinite-horizon ODC problems. The SDP relaxations developed in this work are exact for the design of a centralized controller, hence serving as an alternative for solving Riccati equations. The efficacy of the proposed SDP relaxations is elucidated through a case study on the distributed frequency control of power systems.

Promises of Conic Relaxation for Contingency-Constrained Optimal Power Flow Problem

This paper is concerned with the security-constrained optimal power flow (SCOPF) problem, where each contingency corresponds to the outage of an arbitrary number of lines and generators. The problem is studied by means of a convex relaxation, named semidefinite program (SDP). The existence of a rank-1 SDP solution guarantees the recovery of a global solution of SCOPF. We prove that the rank of the SDP solution is upper bounded by the treewidth of the power network plus one, which is perceived to be small in practice. We then propose a decomposition method to reduce the computational complexity of the relaxation. In the case where the relaxation is not exact, we develop a graph-theoretic convex program to identify the problematic lines of the network and incorporate the loss over those lines into the objective as a penalization (regularization) term, leading to a penalized SDP problem. We perform several simulations on large-scale benchmark systems and verify that the global minima are at most 1% away from the feasible solutions obtained from the proposed penalized relaxation.

Optimal Tap Setting of Voltage Regulation Transformers in Unbalanced Distribution Systems

In this paper, we propose a method to optimally set the tap position of voltage regulation transformers in distribution systems. We cast the problem as a rank-constrained semidefinite program (SDP), in which the transformer tap ratios are captured by 1) introducing a secondary-side “virtual” bus per transformer, and 2) constraining the values that these virtual bus voltages can take according to the limits on the tap positions. Then, by relaxing the non-convex rank-1 constraint in the rank-constrained SDP formulation, one obtains a convex SDP problem. The tap positions are determined as the ratio between the primary-side bus voltage and the secondary-side virtual bus voltage that result from the optimal solution of the relaxed SDP, and then rounded to the nearest discrete tap values. To efficiently solve the relaxed SDP, we propose a distributed algorithm based on the alternating direction method of multipliers (ADMM). We present several case studies with single- and three-phase distribution systems to demonstrate the effectiveness of the distributed ADMM-based algorithm, and compare its results with centralized solution methods.

A Stochastic Beamformed Amplify-and-Forward Scheme in a Multigroup Multicast MIMO Relay Network with Per-Antenna Power Constraints

In this paper, we consider a two-hop one-way relay network for multigroup multicast transmission between long-distance users, in which the relay is equipped with multiple antennas, while the transmitters and receivers are all with a single antenna. Assuming that the perfect channel state information is available, we study amplify-and-forward (AF) schemes that aim at optimizing the max–min-fair (MMF) rate. We begin by considering the classic beamformed AF (BF-AF) scheme, whose corresponding MMF design problem can be formulated as a rank-constrained fractional semidefinite program (SDP). We show that the gap between the BF-AF rate and the SDR rate associated with an optimal SDP solution is sensitive to the number of users as well as the number of power constraints in the relay system. This reveals that the BF-AF scheme may not be well suited for large-scale systems. We, therefore, propose the stochastic beamformed AF (SBF-AF) schemes, which differ from the BF-AF scheme in that time-varying AF weights are used. We prove that the MMF rates of the proposed SBF-AF schemes are at most 0.8317 bits/s/Hz less than the SDR rate, irrespective of the number of users or power constraints. Thus, SBF-AF can outperform BF-AF especially in large-scale systems. Finally, we present numerical results to demonstrate the viability of our proposed schemes.

A semi-definite relaxation approach for partial discharge source location in transformers

Current issues on localization algorithms based on the time difference of arrival (TDOA) for the partial discharge (PD) source include its sensitivity to time delay error, easy local convergence or divergence, and the large amount of computational load and time. A semi-definite relaxation method for PD source location to solve the time delay positioning equations is proposed in this paper, which using the semi-definite programming problem has the characteristic that can ensure obtained the global optimal solution. The proposed method converts nonlinear time delay equations into an equivalent semi-definite programming (SDP) problem by equivalent transformation and rank-1 relaxation firstly. Then use the interior point algorithm to solve the SDP problem that to obtain a unique global optimum solution, and extract a rank-1 component from the global optimal solution of relaxed SDP. Finally, to serve as a good approximate of the original problem for the PD location. The method was used to localize a measured PD source signal in the laboratory, and the results were compared with those of positioning by the Newton-iterative method. The comparison showed that the method can reduce sensitivity for the time delay error as well as effectively solve TDOA location equations, thereby ensuring that the result is a unique global optimal solution with high positioning efficiency. The localization algorithm problem is solved with the unavoidable and difficult-to-locate time delay error.

Layout Decomposition for Triple Patterning Lithography

As minimum feature size and pitch spacing further scale down, triple patterning lithography is a likely 193 nm extension along the paradigm of double patterning lithography for 14-nm technology node. Layout decomposition, which divides input layout into several masks to minimize the conflict and stitch numbers, is a crucial design step for double/triple patterning lithography. In this paper, we present a systematic study on triple patterning layout decomposition problem, which is shown to be NP-hard. Because of the NP-hardness, the runtime required to exactly solve it increases dramatically with the problem size. We first propose a set of graph division techniques to reduce the problem size. Then, we develop integer linear programming (ILP) to solve it. For large layouts, even with the graph-division techniques, ILP may still suffer from serious runtime overhead. To achieve better trade-off between runtime and performance, we present a novel semidefinite programming (SDP)-based algorithm. Followed by a mapping process, we can translate the SDP solutions into the final decomposition solutions. Experimental results show that the graph division can reduce runtime dramatically. In addition, SDP-based algorithm can achieve great speed-up even compared with accelerated ILP, with very comparable results in terms of the stitch number and the conflict number.

Enhancing Pure-Pixel Identification Performance via Preconditioning

In this paper, we analyze different preconditionings designed to enhance robustness of pure-pixel search algorithms, which are used for blind hyperspectral unmixing and which are equivalent to near-separable nonnegative matrix factorization algorithms. Our analysis focuses on the successive projection algorithm (SPA), a simple, efficient, and provably robust algorithm. Recently, a provably robust preconditioning was proposed by Gillis and Vavasis [SIAM J. Optim., 25 (2015), pp. 677--698] which requires the resolution of a semidefinite program (SDP). Since solving the SDP in high precisions can be time consuming, we generalize the robustness analysis to approximate solutions of the SDP showing that a high accuracy solution is not crucial for robustness, paving the way for faster preconditionings. This first contribution also allows us to provide a robustness analysis for two other preconditionings. The first one is prewhitening, which can be interpreted as an optimal solution of the same SDP with additional constraints. We analyze the robustness of prewhitening, which allows us to characterize situations in which it performs competitively with the SDP-based preconditioning. The second one is based on SPA itself and can be interpreted as an optimal solution of a relaxation of the SDP. It is extremely fast when competing with the SDP-based preconditioning on several synthetic data sets.

Convex Relaxation for Optimal Power Flow Problem: Mesh Networks

This paper is concerned with the optimal power flow (OPF) problem. We have recently shown that a convex relaxation based on semidefinite programming (SDP) is able to find a global solution of OPF for IEEE benchmark systems, and moreover this technique is guaranteed to work over acyclic (distribution) networks. The present work studies the potential of the SDP relaxation for OPF over mesh (transmission) networks. First, we consider a simple class of cyclic systems, namely weakly-cyclic networks with cycles of size 3. We show that the success of the SDP relaxation depends on how the line capacities are modeled mathematically. More precisely, the SDP relaxation is proven to succeed if the capacity of each line is modeled in terms of bus voltage difference, as opposed to line active power, apparent power or angle difference. This result elucidates the role of the problem formulation. Our second contribution is to relate the rank of the minimum-rank solution of the SDP relaxation to the network topology. The goal is to understand how the computational complexity of OPF is related to the underlying topology of the power network. To this end, an upper bound is derived on the rank of the SDP solution, which is expected to be small in practice. A penalization method is then applied to the SDP relaxation to enforce the rank of its solution to become 1, leading to a near-optimal solution for OPF with a guaranteed optimality degree. The remarkable performance of this technique is demonstrated on IEEE systems with more than 7000 different cost functions.

A Gaussian process model for data association and a semidefinite programming solution.

In this paper, we propose a Bayesian model for the data association problem, in which trajectory smoothness is enforced through the use of Gaussian process priors. This model allows to score candidate associations using the evidence framework, thus casting the data association problem into an optimization problem. Under some additional mild assumptions, this optimization problem is shown to be equivalent to a constrained Max K-section problem. Furthermore, for K=2, a MaxCut formulation is obtained, to which an approximate solution can be efficiently found using an SDP relaxation. Solving this MaxCut problem is equivalent to finding the optimal association out of the combinatorially many possibilities. The obtained clustering depends only on two hyperparameters, which can also be selected by maximum evidence.

Safe Bounds in Semidefinite Programming by Using Interval Arithmetic

Efficient solvers for optimization problems are based on linear and semidefinite relaxations that use floating point arithmetic. However, due to the rounding errors, relaxation thus may overestimate, or worst, underestimate the very global optima. The purpose of this article is to introduce an efficient and safe procedure to rigorously bound the global optima of semidefinite program. This work shows how, using interval arithmetic, rigorous error bounds for the optimal value can be computed by carefully post processing the output of a semidefinite programming solver. A lower bound is computed on a semidefinite relaxation of the constraint system and the objective function. Numerical results are presented using the SDPA (SemiDefinite Programming Algorithm), solver to compute the solution of semidefinite programs. This rigorous bound is injected in a branch and bound algorithm to solve the optimisation problem.

Guaranteed clustering and biclustering via semidefinite programming

Identifying clusters of similar objects in data plays a significant role in a wide range of applications. As a model problem for clustering, we consider the densest $$k$$ -disjoint-clique problem, whose goal is to identify the collection of $$k$$ disjoint cliques of a given weighted complete graph maximizing the sum of the densities of the complete subgraphs induced by these cliques. In this paper, we establish conditions ensuring exact recovery of the densest $$k$$ cliques of a given graph from the optimal solution of a particular semidefinite program. In particular, the semidefinite relaxation is exact for input graphs corresponding to data consisting of $$k$$ large, distinct clusters and a smaller number of outliers. This approach also yields a semidefinite relaxation with similar recovery guarantees for the biclustering problem. Given a set of objects and a set of features exhibited by these objects, biclustering seeks to simultaneously group the objects and features according to their expression levels. This problem may be posed as that of partitioning the nodes of a weighted bipartite complete graph such that the sum of the densities of the resulting bipartite complete subgraphs is maximized. As in our analysis of the densest $$k$$ -disjoint-clique problem, we show that the correct partition of the objects and features can be recovered from the optimal solution of a semidefinite program in the case that the given data consists of several disjoint sets of objects exhibiting similar features. Empirical evidence from numerical experiments supporting these theoretical guarantees is also provided.

Global Optimization of Non-Convex Hydro-Thermal Coordination Based on Semidefinite Programming

In order to reduce the fuel cost of thermal units, a semidefinite programming (SDP) method is used to solve a hydrothermal coordination (HTC) optimization problem. By manipulating the structure of decision variable matrix, the original nonconvex problem is reformulated into a convex SDP relaxation model without sacrificing the nonlinear relation among hydropower generation, reservoir storage volume, and discharge water. A global minimum is therefore guaranteed and well-developed convex optimization theories can thus be employed to solve the problem. Both sparse matrix techniques and a simplified SDP model are discussed to reduce computational cost. One mostly used HTC case is employed to test the performance of the proposed method. Detailed comparisons between the proposed and other methods show that the final result of SDP model is by far the best result ever. In addition, a large-sized HTC case shows the potential of SDP in practical use. Finally, we prove that as a relaxation technique, the SDP solution, which satisfies all the constraints, is indeed the optimal solution of the original nonconvex problem.

New Semidefinite Relaxation Method for Acoustic Energy-Based Source Localization

The acoustic energy-based source localization problem is addressed. Based on the noisy acoustic energy measurements, we obtain a nonlinear and non-convex weighted least squares (WLS) formulation for this problem, whose globally optimal solution is hardly obtained without a good initial estimate. To overcome this difficulty, we first transform the original measurement model to obtain an approximate WLS formulation, and then employ the semidefinite relaxation (SDR) technique to obtain a semidefinite programming (SDP), which can be solved very efficiently. The SDP solution is further refined using the conventional Gaussian randomization procedure. Moreover, we propose an alternating estimation procedure to handle the localization problem when the decay factor is unknown. Simulation results show that the proposed SDR method significantly outperforms the existing methods.

High-accuracy solution of large-scale semidefinite programs

We present a first-order approach for solving semidefinite programs. The goal of this approach is to compute a solution of the semidefinite program (SDP) up to a high accuracy in spite of using only partial second-order information. We propose a hybrid approach that uses an accelerated projection method to generate an approximate solution and then switches to the quasiminimal residual algorithm (QMR) algorithm applied to a symmetrized version of the Alizadeh-Haeberly-Overton (AHO) system to improve this approximation. Some numerical experiments based on a number of random test examples illustrate the potential of this approach.

The tracial moment problem and trace-optimization of polynomials

The main topic addressed in this paper is trace-optimization of polynomials in noncommuting (nc) variables: given an nc polynomial f, what is the smallest trace \({f(\underline {A})}\) can attain for a tuple of matrices \({\underline {A}}\)? A relaxation using semidefinite programming (SDP) based on sums of hermitian squares and commutators is proposed. While this relaxation is not always exact, it gives effectively computable bounds on the optima. To test for exactness, the solution of the dual SDP is investigated. If it satisfies a certain condition called flatness, then the relaxation is exact. In this case it is shown how to extract global trace-optimizers with a procedure based on two ingredients. The first is the solution to the truncated tracial moment problem, and the other crucial component is the numerical implementation of the Artin-Wedderburn theorem for matrix *-algebras due to Murota, Kanno, Kojima, Kojima, and Maehara. Trace-optimization of nc polynomials is a nontrivial extension of polynomial optimization in commuting variables on one side and eigenvalue optimization of nc polynomials on the other side—two topics with many applications, the most prominent being to linear systems engineering and quantum physics. The optimization problems discussed here facilitate new possibilities for applications, e.g. in operator algebras and statistical physics.