Convex Formulation Research Articles

Risk-coherent H2-optimal disturbance rejection under model uncertainty

A control design procedure for disturbance rejection, when the disturbance model is uncertain, is proposed. We use the probabilistic information about the process disturbance model to design a controller to account for the uncertainty by using a risk-theoretical approach. By introducing the notion of coherent measures of risk, we analyze standard approaches to account for this uncertainty, and we intend to show that the conditional value-at-risk (CVaR) is an appropriate function to measure the uncertainty in the disturbance model. We also derive a convex formulation for the controller design problem when the Youla parameter is linearly parametrized. A numerical example illustrates the main discussion of this article.

A Method for Identification of Markovian Jump ARX Processes

In this paper, we address the problem of identification of Markovian jump ARX systems in the case where there is no a priori knowledge on the statistics of the process noise. The only assumptions made are the availability of known upper bounds on the subsystems order and noise level. The proposed method leverages available results to identify the subsystems models and provides a novel way to estimate the transition probability matrix of the switching in the presence of uncertainty. More precisely, since one cannot observe the exact switching sequence, the problem of estimating the transition probability matrix is formulated as a robust optimization problem, where one maximizes the minimum of the probability of all possible switching sequences. We show that this robust optimization problem can be efficiently solved by providing an equivalent convex formulation that can be solved using off-the-shelf software. Several academic examples are provided which illustrate the effectiveness of proposed approach.

Convex Formulation for Kernel PCA and Its Use in Semisupervised Learning.

In this brief, kernel principal component analysis (KPCA) is reinterpreted as the solution to a convex optimization problem. Actually, there is a constrained convex problem for each principal component, so that the constraints guarantee that the principal component is indeed a solution, and not a mere saddle point. Although these insights do not imply any algorithmic improvement, they can be used to further understand the method, formulate possible extensions, and properly address them. As an example, a new convex optimization problem for semisupervised classification is proposed, which seems particularly well suited whenever the number of known labels is small. Our formulation resembles a least squares support vector machine problem with a regularization parameter multiplied by a negative sign, combined with a variational principle for KPCA. Our primal optimization principle for semisupervised learning is solved in terms of the Lagrange multipliers. Numerical experiments in several classification tasks illustrate the performance of the proposed model in problems with only a few labeled data.

Fast Hyperspectral Unmixing in Presence of Nonlinearity or Mismodeling Effects

This paper presents two novel hyperspectral mixture models and associated unmixing algorithms. The two models assume a linear mixing model corrupted by an additive term whose expression can be adapted to account for multiple scattering nonlinearities (NL), or mismodeling effects (ME). The NL model generalizes bilinear models by taking into account higher order interaction terms. The ME model accounts for different effects, such as endmember variability or the presence of outliers. The abundance and residual parameters of these models are estimated by considering a convex formulation suitable for fast estimation algorithms. This formulation accounts for constraints, such as the sum-to-one and nonnegativity of the abundances, the nonnegativity of the nonlinearity coefficients, the spectral smoothness of the ME terms and the spatial sparseness of the residuals. The resulting convex problem is solved using the alternating direction method of multipliers whose convergence is ensured theoretically. The proposed mixture models and their unmixing algorithms are validated on both synthetic and real images showing competitive results regarding the quality of the inference and the computational complexity when compared to the state-of-the-art algorithms.

Rank-constrained optimization and its applications

This paper investigates an iterative approach to solve the general rank-constrained optimization problems (RCOPs) defined to optimize a convex objective function subject to a set of convex constraints and rank constraints on unknown rectangular matrices. In addition, rank minimization problems (RMPs) are introduced and equivalently transformed into RCOPs by introducing a quadratic matrix equality constraint. The rank function is discontinuous and nonconvex, thus the general RCOPs are classified as NP-hard in most of the cases. An iterative rank minimization (IRM) method, with convex formulation at each iteration, is proposed to gradually approach the constrained rank. The proposed IRM method aims at solving RCOPs with rank inequalities constrained by upper or lower bounds, as well as rank equality constraints. Proof of the convergence to a local minimizer with at least a sublinear convergence rate is provided. Four representative applications of RCOPs and RMPs, including system identification, output feedback stabilization, and structured H2 controller design problems, are presented with comparative simulation results to verify the feasibility and improved performance of the proposed IRM method.

Two-echelon supply chain coordination under information asymmetry with multiple types

We analyse a principal-agent contracting model with asymmetric information between a supplier and a retailer. Both the supplier and the retailer have the classical non-linear economic ordering cost functions consisting of ordering and holding costs. We assume that the retailer has the market power to enforce any order quantity. Furthermore, the retailer has private holding costs. The supplier wants to minimise his expected costs by offering a menu of contracts with side payments as an incentive mechanism. We consider a general number of discrete single-dimensional retailer types with type-dependent default options. A natural and common model formulation is non-convex, but we present an equivalent convex formulation. Hence, the contracting model can be solved efficiently for a general number of retailer types. We also derive structural properties of the optimal menu of contracts. In particular, we completely characterise the optimum for two retailer types and provide a minimal list of candidate contracts for three types. We show that the retailer’s lying behaviour is more complex than simply lying to have higher costs. Finally, we prove a sufficient condition to guarantee unique contracts in the optimal solution for a general number of retailer types.

Comparative Analysis of Unipolar and Bipolar Control of Modular Battery for Thermal and State-of-Charge Balancing

Thermal and state-of-charge (SOC) imbalance is a well-known issue that causes nonuniform aging in batteries. The modular battery based on cascaded converters is a potential solution to this problem. This paper presents bipolar control (BPC) of a modular battery and compares it with previously proposed unipolar control (UPC) mode in terms of thermal/SOC balancing performance and energy efficiency. The BPC needs four-quadrant operation of a full-bridge (FB) converter using bipolar pulsewidth modulation (PWM) inside each module, whereas UPC only needs a half-bridge (HB) converter with unipolar PWM. The BPC, unlike UPC, enables charging of some cells while discharging others. An averaged state-space electrothermal battery model is derived for a convex formulation of the balancing control problem. The control problem is formulated on a constrained linear quadratic (LQ) form and solved in a model predictive control (MPC) framework using one-step-ahead prediction. The simulation results show that BPC, without even requiring load current variations, gives better balancing performance than UPC but at the cost of reduced efficiency. The UPC requires at least current direction reversal for acceptable balancing performance. In short, the UPC is a more cost- and energy-efficient solution for electric vehicle (EV) and plug-in hybrid EV applications, whereas the BPC can be beneficial in applications involving load cycles with high current pulses of long duration.

ML2Motif-Reliable extraction of discriminative sequence motifs from learning machines.

High prediction accuracies are not the only objective to consider when solving problems using machine learning. Instead, particular scientific applications require some explanation of the learned prediction function. For computational biology, positional oligomer importance matrices (POIMs) have been successfully applied to explain the decision of support vector machines (SVMs) using weighted-degree (WD) kernels. To extract relevant biological motifs from POIMs, the motifPOIM method has been devised and showed promising results on real-world data. Our contribution in this paper is twofold: as an extension to POIMs, we propose gPOIM, a general measure of feature importance for arbitrary learning machines and feature sets (including, but not limited to, SVMs and CNNs) and devise a sampling strategy for efficient computation. As a second contribution, we derive a convex formulation of motifPOIMs that leads to more reliable motif extraction from gPOIMs. Empirical evaluations confirm the usefulness of our approach on artificially generated data as well as on real-world datasets.

Object Depth Profile and Reflectivity Restoration From Sparse Single-Photon Data Acquired in Underwater Environments

This paper presents two new algorithms for the joint restoration of depth and reflectivity (DR) images constructed from time-correlated single-photon counting measurements. Two extreme cases are considered: 1) a reduced acquisition time that leads to very low photon counts; and 2) imaging in a highly attenuating environment (such as a turbid medium), which makes the reflectivity estimation more difficult at increasing range. Adopting a Bayesian approach, the Poisson distributed observations are combined with prior distributions about the parameters of interest, to build the joint posterior distribution. More precisely, two Markov random field (MRF) priors enforcing spatial correlations are assigned to the DR images. Under some justified assumptions, the restoration problem (regularized likelihood) reduces to a convex formulation with respect to each of the parameters of interest. This problem is first solved using an adaptive Markov chain Monte Carlo (MCMC) algorithm that approximates the minimum mean square parameter estimators. This algorithm is fully automatic since it adjusts the parameters of the MRFs by maximum marginal likelihood estimation. However, the MCMC-based algorithm exhibits a relatively long computational time. The second algorithm deals with this issue and is based on a coordinate descent algorithm. Results on single-photon depth data from laboratory-based underwater measurements demonstrate the benefit of the proposed strategy that improves the quality of the estimated DR images.

Mixed Discrete-Continuous Planning with Convex Optimization

Robots operating in the real world must be able to handle both discrete and continuous change. Many robot behaviors can be controlled through numeric parameters (called control variables), which affect the rate of the continuous change. Previous approaches capable of reasoning efficiently with control variables impose severe restrictions that limit the expressivity of the problems that can be solved. A broad class of robotic applications require, for example, convex quadratic constraints on state variables and control variables that are jointly constrained and that affect multiple state variables simultaneously. However, extensions to prior approaches are not straightforward, since these characteristics are non-linear and hard to scale. We introduce cqScotty, a heuristic forward search planner that solves these problems efficiently. While naive formulations of consistency checks are not convex and do not scale, cqScotty uses an efficient convex formulation, in the form of a Second Order Cone Program (SOCP), that is very fast to solve. We demonstrate the scalability of our approach on three new realistic domains.

Joint Estimation of Multiple Conditional Gaussian Graphical Models.

In this paper, we propose a joint conditional graphical Lasso to learn multiple conditional Gaussian graphical models, also known as Gaussian conditional random fields, with some similar structures. Our model builds on the maximum likelihood method with the convex sparse group Lasso penalty. Moreover, our model is able to model multiple multivariate linear regressions with unknown noise covariances via a convex formulation. In addition, we develop an efficient approximated Newton's method for optimizing our model. Theoretically, we establish the asymptotic properties of our model on consistency and sparsistency under the high-dimensional settings. Finally, extensive numerical results on simulations and real data sets demonstrate that our method outperforms the compared methods on structure recovery and structured output prediction. To the best of our knowledge, the joint learning of multiple multivariate regressions with unknown covariance is first studied.

Joint Optimal Pricing and Electrical Efficiency Enforcement for Rational Agents in Microgrids

In electrical distribution grids, the constantly increasing number of power generation devices based on renewables demands a transition from a centralized to a distributed generation paradigm. In fact, power injection from distributed energy resources (DERs) can be selectively controlled to achieve other objectives beyond supporting loads, such as the minimization of the power losses along the distribution lines and the subsequent increase of the grid hosting capacity. However, these technical achievements are only possible if alongside electrical optimization schemes, a suitable market model is set up to promote cooperation from the end users. In contrast with the existing literature, where energy trading and electrical optimization of the grid are often treated separately, or the trading strategy is tailored to a specific electrical optimization objective, in this paper, we consider their joint optimization. We also allow for a modular approach, where the market model can support any smart grid optimization goal. Specifically, we present a multi-objective optimization problem accounting for energy trading, where: 1) DERs try to maximize their profit, resulting from selling their surplus energy; 2) the loads try to minimize their expense; and 3) the main power supplier aims at maximizing the electrical grid efficiency through a suitable discount policy. This optimization problem is proved to be non-convex, and an equivalent convex formulation is derived. Centralized solutions are discussed and a procedure to distribute the solution is proposed. Numerical results to demonstrate the effectiveness of the so obtained optimal policies are finally presented, showing the proposed model results in economic benefits for all the users (generators and loads) and in an increased electrical efficiency for the grid.

Simultaneous Contact, Gait and Motion Planning for Robust Multi-Legged Locomotion via Mixed-Integer Convex Optimization

Traditional motion planning approaches for multi-legged locomotion divide the problem into several stages, such as contact search and trajectory generation. However, reasoning about contacts and motions simultaneously is crucial for the generation of complex whole-body behaviors. Currently, coupling theses problems has required either the assumption of a fixed gait sequence and flat terrain condition, or non-convex optimization with intractable computation time. In this paper, we propose a mixed-integer convex formulation to plan simultaneously contact locations, gait transitions and motion, in a computationally efficient fashion. In contrast to previous works, our approach is not limited to flat terrain nor to a pre-specified gait sequence. Instead, we incorporate the friction cone stability margin, approximate the robot's torque limits, and plan the gait using mixed-integer convex constraints. We experimentally validated our approach on the HyQ robot by traversing different challenging terrains, where non-convexity and flat terrain assumptions might lead to sub-optimal or unstable plans. Our method increases the motion generality while keeping a low computation time.

A Convex Formulation of Traffic Dynamics on Transportation Networks

This article proposes a numerical scheme for computing the evolution of vehicular traffic on a road network over a finite time horizon. The traffic dynamics on each link is modeled by the Hamilton--Jacobi (HJ) partial differential equation (PDE), which is an equivalent form of the Lighthill--Whitham--Richards PDE. The main contribution of this article is the construction of a single convex optimization program which computes the traffic flow at a junction over a finite time horizon and decouples the PDEs on connecting links. Compared to discretization schemes which require the computation of all traffic states on a time-space grid, the proposed convex optimization approach computes the boundary flows at the junction using only the initial condition on links and the boundary conditions of the network. The computed boundary flows at the junction specify the boundary condition for the HJ PDE on connecting links, which then can be separately solved using an existing semi-explicit scheme for single link HJ PDE...

Formulating the Torsion Strength of Dry-Stacked Stone Blocks by Comparing Convex and Concave Contact Formulations and Experimental Results

Objectives: Objective of this work is to investigate the reliability of the two formulations most commonly used in modeling the geometric contact of dry-stacked stone blocks, i.e., those based on concave and convex contact assumptions. Methods: A comparison between these two formulations for the yield domains of torsion and torsion-shear interaction is carried out; the analytical results are then compared with experimental outcomes existing in the literature and difference percentages are evaluated in terms of shear forces. For both formulations, the reference model of two dry-stacked rigid blocks is adopted with the hypotheses of infinite compressive strength for blocks, absence of tensile strength and frictional behaviour at their contact. Findings: The analysis underlines that the convex formulation provides more reliable results in terms of both pure torsion and interaction between torsion and shear. In fact, the percentage difference between experimental and analytical shear forces results to be very small in this case, while a great difference is estimated for concave formulation, both for pure torsion and torsion-shear interaction. Then a possible correction of the torsion capacity in the concavity model is proposed assuming a proper reduction of the contact area. This simple criterion allows obtaining a good agreement of the yield domains with those obtained by convexity model and experimental results. Application/Improvements: With the proposed correction the simplicity of the algorithms of the concave formulation can find interesting applications in rigid block limit analysis, especially for non-associative friction solutions of 3D masonry block assemblages, thanks to the very low computational effort compared with all the other existing solution procedures.Additional experimental work is being carried out to fully validate such correction and to investigate its effect on the torsion-bending moment interaction.

Convexification of the Nash Bargaining Based Environmental-Economic Dispatch

The environmental-economic dispatch (EED) is a bi-objective optimization problem. The Nash bargaining theory provides one way to determine a compromising solution without a clear carbon tax or carbon cap, or any subjective attitudes on both objectives, which yields a non-convex program. This letter proposes a convex formulation for Nash bargaining based EED and a linear programming (LP) based algorithm. Case studies show its scalability on real-world large-scale power systems.

A Rational Convex Program for Linear Arrow-Debreu Markets

We present a new flow-type convex program describing equilibrium solutions to linear Arrow-Debreu markets. Whereas convex formulations were previously known ([Nenakov and Primak 1983; Jain 2007; Cornet 1989]), our program exhibits several new features. It provides a simple necessary and sufficient condition and a concise proof of the existence and rationality of equilibria, settling an open question raised by Vazirani [2012]. As a consequence, we also obtain a simple new proof of the result in Mertens [2003] that the equilibrium prices form a convex polyhedral set.

Localization From Connectivity: A 1-bit Maximum Likelihood Approach

We consider the problem of determining the location of sensor nodes in a wireless sensor ad hoc network when only connectivity information is available, i.e., one only knows if a pair of nodes is within a fixed radio range (distance) of each other but does not have access to exact or even approximate distance information. We propose a maximum likelihood based reconstruction algorithm to reconstruct the node positions in a $d$ -dimensional Euclidean space. For an $n$ -node network, the constrained maximum likelihood estimation problem is non-convex in both the $d \times n$ node position matrix $X$ and in the Gram matrix $Q_{X}=X^{\top} X$ . We derive an upperbound on the average Frobenius norm of the estimation error in $Q$ , which is of the order of the reciprocal of square root of $n$ for a fixed radio range. We present a set of algorithms for finding the maximum likelihood estimate of $X$ by first embedding $d \times n~X$ into $m \times n~Y$ , factorizing $Q_{Y} = Y^{\top} Y$ , $d \leq m \leq n$ , and then optimizing in $Y$ . We relate local minima in $Y$ to the global minimum of a relaxed convex formulation in $Q_{Y}$ to provide global convergence guarantees, despite the nonconvexity of the negative log-likelihood function in $Y$ . We demonstrate that our algorithm is empirically successful for both uniform and irregular networks, using only a few anchor nodes. Finally, numerical experiments demonstrate improved performance of the proposed algorithm relative to the MDS algorithm and a variant.

Optimal management of unbalanced smart microgrids for scheduled and unscheduled multiple transitions between grid-connected and islanded modes

The paper proposes a novel optimization framework for smart microgrids for both scheduled and unscheduled multiple transitions between grid-connected and isolated modes. The optimization aims at maximizing the overall microgrid profit, from selling energy to grid, and minimizing consumption costs by scheduling dispatchable loads and generation, energy storages, and load curtailment while satisfying all network and operational constraints. For unscheduled transitions, the time of mode transitions in the planning period is not known in advance. So, the optimization is reinitialized and restarted online after considering the most updated state. Network constraints for three-phase unbalanced distribution microgrid model are adopted in details in convex formulation. Bidirectional energy trading with the grid is considered. Moreover, Model Predictive Control (MPC) is used for handling uncertainty in renewable generation and load. Comprehensive simulation results are given for validating the proposed algorithm and comparison is made between results of scheduled and unscheduled optimization.

Detection of faults in rotating machinery using periodic time-frequency sparsity

This paper addresses the problem of extracting periodic oscillatory features in vibration signals for detecting faults in rotating machinery. To extract the feature, we propose an approach in the short-time Fourier transform (STFT) domain where the periodic oscillatory feature manifests itself as a relatively sparse grid. To estimate the sparse grid, we formulate an optimization problem using customized binary weights in the regularizer, where the weights are formulated to promote periodicity. In order to solve the proposed optimization problem, we develop an algorithm called augmented Lagrangian majorization–minimization algorithm, which combines the split augmented Lagrangian shrinkage algorithm (SALSA) with majorization–minimization (MM), and is guaranteed to converge for both convex and non-convex formulation. As examples, the proposed approach is applied to simulated data, and used as a tool for diagnosing faults in bearings and gearboxes for real data, and compared to some state-of-the-art methods. The results show that the proposed approach can effectively detect and extract the periodical oscillatory features.