Semidefinite Programming Formulation Research Articles

Weighted Algebraic Connectivity Maximization for Optical Satellite Networks

In this paper, the topology configuration methods for heterogeneous optical satellite networks are investigated. Our objectives are to maximize weighted algebraic connectivity with respect to both network initialization and reconfiguration scenarios subject to onboard hardware constraints. The problems are not strictly convex and have been proven as NP-hard. In order to solve the problems in polynomial time, the original problems are relaxed to a convex optimization problem. Specifically, the relaxed problem is transformed to a positive semidefinite programming form, which can be solved exactly and more efficiently. Furthermore, based on the perturbation theory of matrices, we propose two greedy heuristic methods to deal with the initialization and reconfiguration case from the relaxed solutions, respectively. Simulation results show that the proposed algorithms are able to accomplish network initialization and reconfiguration correctly in the overwhelming majority of situations. The final sub-optimal solutions can also be obtained under low computational complexity.

Lieb's concavity theorem, matrix geometric means, and semidefinite optimization

A famous result of Lieb establishes that the map (A,B)↦tr[K⁎A1−tKBt] is jointly concave in the pair (A,B) of positive definite matrices, where K is a fixed matrix and t∈[0,1]. In this paper we show that Lieb's function admits an explicit semidefinite programming formulation for any rational t∈[0,1]. Our construction makes use of a semidefinite formulation of weighted matrix geometric means. We provide an implementation of our constructions in Matlab.

Optimizing the convergence rate of the quantum consensus: A discrete-time model

Motivated by the recent advances in the field of quantum computing, quantum systems are modeled and analyzed as networks of decentralized quantum nodes which employ distributed quantum consensus algorithms for coordination. In the literature, both continuous and discrete-time models of the algorithm have been proposed. This paper aims at optimizing the convergence rate of the discrete-time quantum consensus algorithm over a quantum network with N qudits. The induced graphs are categorized in terms of the partitions of integer N by arranging them as the Schreier graphs. It is shown that the original optimization problem reduces to optimizing the Second Largest Eigenvalue Modulus of the weight matrix. Exploiting the Specht module representation of partitions of N, the Aldous’ conjecture is generalized to all partitions of integer N implying that the spectral gap of all resultant induced graphs is the same. The spectral radius of the Laplacian is obtained from the feasible least dominant partition in the Hasse diagram of integer N. By analytically addressing the semidefinite programming formulation of the problem, closed-form expressions for the optimal results are provided for a wide range of topologies.

Covariance Matrix Estimation for Interest-Rate Risk Modeling via Smooth and Monotone Regularization

Estimating covariance matrices in high-dimensional settings is a challenging problem central to modern finance. The sample covariance matrix is well-known to give poor estimates in high dimensions with insufficient samples, and may cause severe risk underestimates of optimized portfolios in the Markowitz framework. In order to provide useful estimates in this regime, a variety of improved covariance matrix estimates have been developed that exploit additional structure in the data. Popular approaches include low-rank (principal component and factor analysis) models, banded structure, sparse inverse covariances, and parametric models. We investigate a novel nonparametric prior for random vectors which have a spatial ordering: we assume that the covariance is monotone and smooth with respect to this ordering. This applies naturally to problems such as interest-rate risk modeling, where correlations decay for contracts that are further apart in terms of expiration dates. We propose a convex optimization (semi-definite programming) formulation for this estimation problem, and develop efficient algorithms. We apply our framework for risk measurement and forecasting with Eurodollar futures, investigate limited, missing and asynchronous data, and show that it provides valid (positive-definite) covariance estimates more accurate than existing methods.

On the superlinear local convergence of a penalty-free method for nonlinear semidefinite programming

This paper is concerned with a sequentially semidefinite programming (SSDP) algorithm for solving nonlinear semidefinite programming problems (NLSDP), which does not use a penalty function or a filter. This method, inspired by the classic SQP method, calculates a trial step by a quadratic semidefinite programming subproblem at each iteration. The trial step is determined such that either the value of the objective function or the measure of constraint violation is sufficiently reduced. In order to guarantee global convergence, the measure of constraint violation in each iteration is required not to exceed a progressively decreasing limit. We prove the global convergence properties of the algorithm under mild assumptions. We also analyze the local behaviour of the proposed method while using a second order correction strategy to avoid Maratos effect. We prove that, under the strict complementarity and the strong second order sufficient conditions with the sigma term, the rate of local convergence is superlinear. Finally, some numerical results with nonlinear semidefinite programming formulation of control design problem with the data contained in COMPleib are given.

Large-Scale Binary Quadratic Optimization Using Semidefinite Relaxation and Applications.

In computer vision, many problems can be formulated as binary quadratic programs (BQPs), which are in general NP hard. Finding a solution when the problem is of large size to be of practical interest typically requires relaxation. Semidefinite relaxation usually yields tight bounds, but its computational complexity is high. In this work, we present a semidefinite programming (SDP) formulation for BQPs, with two desirable properties. First, it produces similar bounds to the standard SDP formulation. Second, compared with the conventional SDP formulation, the proposed SDP formulation leads to a considerably more efficient and scalable dual optimization approach. We then propose two solvers, namely, quasi-Newton and smoothing Newton methods, for the simplified dual problem. Both of them are significantly more efficient than standard interior-point methods. Empirically the smoothing Newton solver is faster than the quasi-Newton solver for dense or medium-sized problems, while the quasi-Newton solver is preferable for large sparse/structured problems.

Robust Masked Beamforming for MISO Cognitive Radio Networks With Unknown Eavesdroppers

This paper studies a cognitive radio network (CRN), in which a multiple-input single-output (MISO) secondary transmitter (SU-Tx) aims to send confidential messages to its receiver (SU-Rx) in the presence of unknown eavesdroppers, while having to control its generated interference to the primary users (PUs) under a given threshold. Because the eavesdroppers are unknown, this paper considers the artificial noise (AN) approach or masked beamforming to provide the information secrecy. The objective of this paper is to maximize the power of AN in order to degrade the eavesdroppers, subject to the signal-to-interference-plus-noise ratio (SINR) constraint at the SU-Rx, as well as the interference temperature limits (ITLs) for the PUs. In the case of perfect channel state information (CSI), we reveal that the optimal strategy for the information-bearing signal is beamforming. Imperfect CSI cases of bounded and stochastic uncertainties are investigated. In the case of ellipsoid-bounded errors, we derive the equivalent forms for the SINR and ITL constraints and then transform the optimization problem into a form of semidefinite programming (SDP). For the case of probabilistic CSI uncertainties, we propose an outage-constrained robust formulation where the CSI errors are Gaussian distributed. With the aid of two kinds of Bernstein-type inequalities, we reexpress the probabilistic constraints into deterministic forms, which results in a safe approximate solution.

Chance Constrained Robust Downlink Beamforming in Multicell Networks

We introduce a downlink robust optimization approach that minimizes a combination of total transmit power by a multiple antenna base station (BS) within a cell and the resulting aggregate inter-cell interference (ICI) power on the users of the other cells. This optimization is constrained to assure that a set of signal-to-interference-plus-noise ratio (SINR) targets are met at user terminals with certain outage probabilities. The outages are due to the uncertainties that naturally emerge in the estimation of channel covariance matrices between a BS and its intra-cell local users as well as the other users of the other cells. We model these uncertainties using random matrices, analyze their statistical behavior, and formulate a tractable probabilistic approach to the design of optimal robust downlink beamforming vectors. The proposed approach reformulates the original intractable non-convex problem in a semidefinite programming (SDP) form with linear matrix inequality (LMI) constraints. The resulting SDP formulation is convex and numerically tractable under the standard rank relaxation. We compare the proposed chance-constrained approach against two different robust design schemes as well as the worst-case robustness. The simulation results confirm better power efficiency and higher resilience against channel uncertainties of the proposed approach in realistic scenarios.

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.

Model-based optimal design of experiments —Semidefinite and nonlinear programming formulations

We use mathematical programming tools, such as Semidefinite Programming (SDP) and Nonlinear Programming (NLP)-based formulations to find optimal designs for models used in chemistry and chemical engineering. In particular, we employ local design-based setups in linear models and a Bayesian setup in nonlinear models to find optimal designs. In the latter case, Gaussian Quadrature Formulas (GQFs) are used to evaluate the optimality criterion averaged over the prior distribution for the model parameters. Mathematical programming techniques are then applied to solve the optimization problems. Because such methods require the design space be discretized, we also evaluate the impact of the discretization scheme on the generated design. We demonstrate the techniques for finding D-, A- and E-optimal designs using design problems in biochemical engineering and show the method can also be directly applied to tackle additional issues, such as heteroscedasticity in the model. Our results show that the NLP formulation produces highly efficient D-optimal designs but is computationally less efficient than that required for the SDP formulation. The efficiencies of the generated designs from the two methods are generally very close and so we recommend the SDP formulation in practice.

Decentralized Control of Linear Switched Nested Systems With $\ell_{2}$-Induced Norm Performance

This paper considers a decentralized switched control problem where exact conditions for controller synthesis are obtained in the form of semidefinite programming (SDP). The formulation involves a discrete-time switched linear plant that has a nested structure, and whose system matrices switch between a finite number of values according to finite-state automation. The goal of this paper is to synthesize a commensurately nested switched controller to achieve a desired level of $\ell_{2}$ -induced norm performance. The nested structures of both plant and controller are characterized by block lower-triangular system matrices. For this setup, exact conditions are provided for the existence of a finite path-dependent synthesis. These include conditions for the completion of scaling matrices obtained through an extended matrix completion lemma. When individual controller dimensions are chosen at least as large as the plant, these conditions reduce to a set of linear matrix inequalities. The completion lemma also provides an algorithm to complete closed-loop scaling matrices, leading to inequalities for controller synthesis that are solvable either algebraically or numerically through SDP.

Construction and application of an ellipsoidal convex model using a semi-definite programming formulation from measured data

As a set theory-based convex model, the ellipsoidal model provides an attractive framework for treating uncertain-but-bounded variations in the structural reliability analysis and design optimization. However, improper modeling of the uncertainties may give rise to misleading non-probabilistic reliability analysis, thus result in either unsafe or over-conservative designs. This paper presents a systematic study on the mathematical formulation for constructing the minimum-volume ellipsoidal convex model using a given set of sample data, and shows its application in existing methods of non-probabilistic reliability analysis and design optimization of structures with bounded uncertainties. In this method, the uncertain parameters are first divided into groups according to their sources. For each individual group of uncertainties, the minimum-volume ellipsoid problem is reformulated into a semi-definite programming (SDP) problem and thus can be efficiently solved to its global optimum. Further, a linear transformation based on the eigenvalue analysis is employed to map the ellipsoidal model into a standard uncertainty space. This uncertainty modeling technique enables a compact and differentiable bound description of the parameter variations. Moreover, it has another useful property, the affine invariance, which is shown to be necessary for meaningful definition of a non-probabilistic reliability index. The effectiveness and efficiency of the present techniques for convex model construction and the corresponding reliability analysis are demonstrated with numerical examples of structural topology optimization problems with bounded variations arising from different sources.

Semi-definite Programming Approach to Stochastic Combined Heat and Power Environmental/Economic Dispatch Problem

This article presents a solution to the stochastic multi-objective combined heat and power environmental/economic dispatch problem using the semi-definite programming formulation. The vector objective is reduced to an equivalent scalar objective using the weighted sum method. The resulting optimization problem is formulated as a convex optimization via semi-definite programming relaxation. The convex optimization problem was solved to obtain Pareto-optimal solutions. Improvement in the distribution of solution set was achieved through non-linear selection of the weight factor. Simulation was performed on a test problem to investigate the effectiveness of the proposed approach. Results showed that the semi-definite programming based weighted sum method has inherently good convergence property and can have its diversity property improved through weight adaptation.

Semidefinite programming formulation of linear-scaling electronic structure theories

We present a linear-scaling approach based on semidefinite programs (SDPs) to compute the density matrix for effective one-electron theories. Traditional methods constrain the density matrix to represent a Slater determinant and hence rely on parameterization or purification. We eliminate the need for such a constraint by performing an energy minimization over all the convex combinations of density matrices representing Slater determinants. By not relying on purification, the SDP approach not only eliminates accumulation error present in some methods but also reduces the amount of truncation error. Sparsity in the Hamiltonian can be exploited to make the SDP approach scale linearly with system size. Crossovers in computational time with a cubically scaling algorithm are demonstrated for one-dimensional hydrogen chains ranging from ${\text{H}}_{50}$ to ${\text{H}}_{1500}$.

Worst-case topology optimization of self-weight loaded structures using semi-definite programming

The paper concerns worst-case compliance optimization by finding the structural topology with minimum compliance for the loading due to the worst possible acceleration of the structure and attached non-structural masses. A main novelty of the paper is that it is shown how this min-max problem can be formulated as a non-linear semi-definite programming (SDP) problem involving a small-size constraint matrix and how this problem is solved numerically. Our SDP formulation is an extension of an eigenvalue problem seen previously in the literature; however, multiple eigenvalues naturally arise which makes the eigenvalue problem non-smooth, whereas the SDP problem presented in this paper provides a computationally tractable problem. Optimized designs, where the uncertain loading is due to acceleration of applied masses and the weight of the structure itself, are shown in two and three dimensions and we show that these designs satisfy optimality conditions that are also presented.

On the Interference Tolerance of the Primary System in Cognitive Radio Networks

In this letter, we consider underlay spectrum access in a cognitive radio network, where two coexisting cellular systems, i.e., primary and cognitive, share the same frequency band. The throughput performance of this system is directly linked to the tolerated interference level at the primary receiver. We investigate the fundamental trade-off between the interference tolerance in the primary system and the achievable throughput of the cognitive network. We formulate the network performance as a multi-objective optimization problem with a set of objectives including minimizing the interference imposed on each primary user, and maximizing the intended signal received at every secondary user. We then derive an equivalent standard semi-definite programming form and prove that the derived problem always yields rank-one solutions. The obtained solution set provides the best achievable throughput (characterized by the maximum intended signal power at each cognitive user) for a given level of interference tolerance in the primary system. It also gives the achievable gain on the system throughput if the primary system is able to compromise and increase its interference tolerance, such framework could be a base for negotiations between the operators. An improvement of 0.38 bits/s/channel-use is observed on the achievable throughput for 1 dB higher interference tolerance at the primary receivers.

Strong duality in Lasserre’s hierarchy for polynomial optimization

A polynomial optimization problem (POP) consists of minimizing a multivariate real polynomial on a semi-algebraic set \(K\) described by polynomial inequalities and equations. In its full generality it is a non-convex, multi-extremal, difficult global optimization problem. More than an decade ago, J. B. Lasserre proposed to solve POPs by a hierarchy of convex semidefinite programming (SDP) relaxations of increasing size. Each problem in the hierarchy has a primal SDP formulation (a relaxation of a moment problem) and a dual SDP formulation (a sum-of-squares representation of a polynomial Lagrangian of the POP). In this note, we show that there is no duality gap between each primal and dual SDP problem in Lasserre’s hierarchy, provided one of the constraints in the description of set \(K\) is a ball constraint. Our proof uses elementary results on SDP duality, and it does not assume that \(K\) has a strictly feasible point.

Fastest Mixing Reversible Markov Chains on Graphs With Degree Proportional Stationary Distributions

In this technical note, we study two semi-definite programming (SDP) methods of assigning transition probabilities to a Markov chain in order to optimize its mixing rate. In the first SDP formulation, there is a single transition probability parameter to be optimized (the holding probability of vertices) which leads to easier and faster computation as opposed to the more general reversible Markov chain formulation corresponding to a stationary distribution that is proportional to the degree of vertices. By deriving exact analytical results, it is shown that both the single parameter and the degree proportional reversible FMMC formulations tend to yield better results than the symmetric SDP formulation for some well-known graphs.

Stable Camera Motion Estimation Using Convex Programming

We study the inverse problem of estimating $n$ locations $\mathbf{t}_1, \mathbf{t}_2, \ldots, \mathbf{t}_n$ (up to global scale, translation, and negation) in $\mathbb{R}^d$ from noisy measurements of a subset of the (unsigned) pairwise lines that connect them, that is, from noisy measurements of $\pm \frac{\mathbf{t}_i - \mathbf{t}_j}{\|\mathbf{t}_i - \mathbf{t}_j \|_2}$ for some pairs $(i,j)$ (where the signs are unknown). This problem is at the core of the structure from motion (SfM) problem in computer vision, where the $\mathbf{t}_i$ represent camera locations in $\mathbb{R}^3$. The noiseless version of the problem, with exact line measurements, has been considered previously under the general title of parallel rigidity theory, mainly in order to characterize the conditions for unique realization of locations. For noisy pairwise line measurements, current methods tend to produce spurious solutions that are clustered around a few locations. This sensitivity of the location estimates is a well-known problem in SfM, especially for large, irregular collections of images. In this paper we introduce a semidefinite programming (SDP) formulation, specially tailored to overcome the clustering phenomenon. We further identify the implications of parallel rigidity theory for the location estimation problem to be well-posed, and prove exact (in the noiseless case) and stable location recovery results. We also formulate an alternating direction method to solve the resulting semidefinite program, and provide a distributed version of our formulation for large numbers of locations. Specifically for the camera location estimation problem, we formulate a pairwise line estimation method based on robust camera orientation and subspace estimation. Finally, we demonstrate the utility of our algorithm through experiments on real images.

Power System Nonlinear State Estimation Using Distributed Semidefinite Programming

State estimation (SE) is an important task allowing power networks to monitor accurately the underlying system state, which is useful for security-constrained dispatch and power system control. For nonlinear AC power systems, SE amounts to minimizing a weighted least-squares cost that is inherently nonconvex, thus giving rise to many local optima. As a result, estimators used extensively in practice rely on iterative optimization methods, which are destined to return only locally optimal solutions, or even fail to converge. A semidefinite programming (SDP) formulation for SE has been advocated, which relies on the convex semidefinite relaxation (SDR) of the original problem and thereby renders it efficiently solvable. Theoretical analysis under simplified conditions is provided to shed light on the near-optimal performance of the SDR-based SE solution at polynomial complexity. The new approach is further pursued toward complementing traditional nonlinear measurements with linear synchrophasor measurements and reducing computational complexity through distributed implementations. Numerical tests on the standard IEEE 30- and 118-bus systems corroborate that the SE algorithms outperform existing alternatives, and approach near-optimal performance.