Convex Formulation Research Articles

Congestion pricing optimization for shared transport

This project includes the analysis of transportation systems by mathematically modeling the collective decision making of drivers via Nash games. This paper considers two types of traffic pricing models which capture the implication of pricing on shared transport. Both models follow a game theoretic framework and capture the perspective of private shared transport companies that aim to increase their profit while improving social welfare or users’ time cost. The first model was established for users who vary their routing choices but not their transportation modes when the demand levels are fixed. The second model is more appropriate for companies with users who optimize both their routing and choice of transportation based on the pricing policy of the system where demands are elastic. Two pricing policies can be determined by convex formulations of optimal system traffic assignment problems on simple networks with different origins and destinations. The models also show how pricing affects road users’ decisions under different parameter settings. The numerical results confirmed formulation feasibility and gave insight on how the company can have an impact on both their revenue and time cost by modifying their pricing scheme.

Cooperative Output Regulation of Heterogeneous Multiagent Systems: A Global Distributed Control Synthesis Approach

Agent-wise local design methods to synthesize distributed control gains focus on the individual dynamics of each agent to guarantee the overall stability of the system. They are powerful tools due to their scalability. However, the agent-wise local design methods are incapable of maximizing the overall system performance through, for example, decay rate assignment. On the other hand, design methods, which are predicated on a global condition, leads to nonconvex optimization problems. This article considers a global design of an internal model-based distributed dynamic state feedback control law for the linear cooperative output regulation problem. We present a convex formulation of this global design problem based on a structured Lyapunov inequality. Then, the existence of solutions to the structured Lyapunov inequality is investigated. Specifically, we analytically show that the solutions exist for the systems satisfying the agent-wise local sufficient condition. Finally, we compare the proposed method with the agent-wise local design method through numerical examples in terms of conservatism, performance maximization, graph dependency, and scalability.

Randomized Subspace Newton Convex Method Applied to Data-Driven Sensor Selection Problem

The randomized subspace Newton convex methods for the sensor selection problem are proposed. The randomized subspace Newton algorithm is straightforwardly applied to the convex formulation, and the customized method in which the part of the update variables are selected to be the present best sensor candidates is also considered. In the converged solution, almost the same results are obtained by original and randomized-subspace-Newton convex methods. As expected, the randomized-subspace-Newton methods require more computational steps while they reduce the total amount of the computational time because the computational time for one step is significantly reduced by the cubic of the ratio of numbers of randomly updating variables to all the variables. The customized method shows superior performance to the straightforward implementation in terms of the quality of sensors and the computational time.

A Convex Optimization Approach to Dynamic Programming in Continuous State and Action Spaces

In this paper, a convex optimization-based method is proposed for numerically solving dynamic programs in continuous state and action spaces. The key idea is to approximate the output of the Bellman operator at a particular state by the optimal value of a convex program. The approximate Bellman operator has a computational advantage because it involves a convex optimization problem in the case of control-affine systems and convex costs. Using this feature, we propose a simple dynamic programming algorithm to evaluate the approximate value function at pre-specified grid points by solving convex optimization problems in each iteration. We show that the proposed method approximates the optimal value function with a uniform convergence property in the case of convex optimal value functions. We also propose an interpolation-free design method for a control policy, of which performance converges uniformly to the optimum as the grid resolution becomes finer. When a nonlinear control-affine system is considered, the convex optimization approach provides an approximate policy with a provable suboptimality bound. For general cases, the proposed convex formulation of dynamic programming operators can be modified as a nonconvex bi-level program, in which the inner problem is a linear program, without losing uniform convergence properties.

Nonconvex Optimization for Signal Processing and Machine Learning [From the Guest Editors

The articles in this special section focus on nonconvex optimization for signal processing and machine learning. Optimization is now widely recognized as an indispensable tool in signal processing (SP) and machine learning (ML). Indeed, many of the advances in these fields rely crucially on the formulation of suitable optimization models and deployment of efficient numerical optimization algorithms. In the early 2000s, there was a heavy focus on the use of convex optimization techniques to tackle SP and ML applications. This is largely due to the fact that convex optimization problems often possess favorable theoretical and computational properties and that many problems of practical interest have been shown to admit convex formulations or good convex approximations.

Real-Time Control of Plug-in Electric Vehicles for Congestion Management of Radial LV Networks: A Comparison of Implementations

The global proliferation of plug-in electric vehicles (PEVs) poses a major challenge for current and future distribution systems. If uncoordinated, their charging process may cause congestion on both network transformers and feeders, resulting in overheating, deterioration, protection triggering and eventual risk of failure, seriously compromising the stability and reliability of the grid. To mitigate such impacts and increase their hosting capacity in radial distribution systems, the present study compares the levels of effectiveness and performances of three alternative centralized thermal management formulations for a real-time agent-based charge control algorithm that aims to minimize the total impact upon car owners. A linear formulation and a convex formulation of the optimization problem are presented and solved respectively by means of integer linear programming and a genetic algorithm. The obtained results are then compared, in terms of their total impact on the end-users and overall performance, with those of the current heuristic implementation of the algorithm. All implementations were tested using a simulation environment considering multiple vehicle penetration and base load levels, and equipment modeled after commercially available charging stations and vehicles. Results show how faster resolution times are achieved by the heuristic implementation, but no significant differences between formulations exist in terms of network management and end-user impact. Every vehicle reached its maximum charge level while all thermal impacts were mitigated for all considered scenarios. The most demanding scenario showcased over a 30% reduction in the peak load for all thermal variants.

Robust Controller Design for Stochastic Nonlinear Systems via Convex Optimization

This article presents ConVex optimization-based Stochastic steady-state Tracking Error Minimization (CV-STEM), a new state feedback control framework for a class of Itô stochastic nonlinear systems and Lagrangian systems. Its innovation lies in computing the control input by an optimal contraction metric, which greedily minimizes an upper bound of the steady-state mean squared tracking error of the system trajectories. Although the problem of minimizing the bound is nonconvex, its equivalent convex formulation is proposed utilizing SDC parameterizations of the nonlinear system equation. It is shown using stochastic incremental contraction analysis that the CV-STEM provides a sufficient guarantee for exponential boundedness of the error for all time with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\bf \mathcal {L}_2}$</tex-math></inline-formula> -robustness properties. For the sake of its sampling-based implementation, we present discrete-time stochastic contraction analysis with respect to a state- and time-dependent metric along with its explicit connection to continuous-time cases. We validate the superiority of the CV-STEM to PID, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}_\infty$</tex-math></inline-formula> , and baseline nonlinear controllers for spacecraft attitude control and synchronization problems.

Entropy-Based Sensing Schemes for Energy Efficiency in Massive MTC

Machine-type-communications (MTC) are crucial in the evolution of mobile communication systems. Within this context, we distinguish the so-called massive MTC (mMTC), where a large number of devices coexist in the same geographical area. In the case of sensors, a high correlation in the collected information is expected. In this letter, we evaluate the impact of correlation on the entropy of a set of quantized Gaussian sources. This model allows us to express the sensed data with the data correlation matrix. Given the nature of mMTC, these matrices may be well approximated as rank deficient. Accordingly, we exploit this singularity to design a technique for switching off several sensors that maximizes the entropy under power-related constraints. The discrete optimization problem is transformed into a convex formulation that can be solved numerically.

Modal Field Synthesis of Monopulse Difference Patterns for Radiating Aperture

This communication presents a rigorous method to synthesize radiating aperture field distributions that produce monopulse difference patterns. This method is based on the modal expansion of the electromagnetic fields at the aperture, and thanks to the use of a convex formulation, it is guaranteed that the achieved result is the optimal solution for the addressed problem. Besides, the synthesis method is valid for any aperture geometry and desired polarization. First, the theory of the synthesis method is presented, and then, several numerical examples with different aperture geometries and polarizations are shown to demonstrate the capabilities of the proposed technique. Finally, the formulation is generalized to allow the synthesis of radiation patterns with nulls and lobes of arbitrary amplitude and phase in any desired direction.

Maximum Margin Multi-Dimensional Classification

Multi-dimensional classification (MDC) assumes heterogenous class spaces for each example, where class variables from different class spaces characterize semantics of the example along different dimensions. Due to the heterogeneity of class spaces, the major difficulty in designing margin-based MDC techniques lies in that the modeling outputs from different class spaces are not comparable to each other. In this paper, a first attempt towards maximum margin multi-dimensional classification is investigated. Following the one-vs-one decomposition within each class space, the resulting models are optimized by leveraging classification margin maximization on individual class variable and model relationship regularization across class variables. We derive convex formulation for the maximum margin MDC problem, which can be tackled with alternating optimization admitting QP or closed-form solution in either alternating step. Experimental studies over real-world MDC data sets clearly validate effectiveness of the proposed maximum margin MDC techniques.

Cooperative Transport of a Slung Load Using Load-Leading Control

A system for cooperative transport of a slung load by a team of autonomous rotorcraft is described. The method described here is hierarchical and centralized, with the payload defined as the leader (here denoted as load-leading control): the payload uses knowledge of its current state and its desired trajectory to determine the net force and moment acting upon its center of gravity required to follow the desired trajectory. Using knowledge of the cable attachment geometry, the required cable forces are computed and transmitted to the rotorcraft. The cable force optimization problem is defined and shown to be nonconvex; constraints that make the problem convex are introduced. Cable forces computed using both the convex and nonconvex formulations are compared. Controllers for both payload and rotorcraft trajectory tracking are designed. Hardware implementation that uses small single-board computers carried onboard the payload and rotorcraft is described. Flight demonstrations conducted in an indoor motion capture studio using preplanned trajectories as well as human-in-the-loop control of the payload are described to show trajectory following and rejection of impulse disturbances. An outdoor flight test showing coordinated transport in the presence of light winds is described. All calculations are hosted on the computers carried on the payload and rotorcraft, and the system is shown to be capable of real-time control and path following both for fixed and in-flight changing formations.

Optimal energy management for hybrid electric aircraft

A convex formulation is proposed for optimal energy management in aircraft with hybrid propulsion systems consisting of gas turbine and electric motor components. By combining a point-mass aircraft dynamical model with models of electrical and mechanical powertrain losses, the fuel consumed over a planned future flight path is minimised subject to constraints on the battery, electric motor and gas turbine. The resulting optimisation problem is used to define a predictive energy management control law that takes into account the variation in aircraft mass during flight. A simulation study based on a representative 100-seat aircraft with a prototype parallel hybrid electric propulsion system is used to investigate the properties of the controller. We show that an optimisation-based control strategy can provide significant fuel savings over heuristic energy management strategies in this context.

PhaseEqual: Convex Phase Retrieval via Alternating Direction Method of Multipliers

The problem of estimating an unknown complex signal from its magnitude-only measurements is a classical problem known as phase retrieval. This problem arises naturally because in some applications, it is difficult or costly to measure the phase of the measurements. In this paper, we propose a new convex formulation, referred to as PhaseEqual, for solving the phase retrieval problem. Similar to PhaseMax T. Goldstein and C. Studer, 2018 by Goldstein and Studer (IEE TIT, 2018), PhaseEqual works in the natural signal space and thus is computationally efficient. We then extend PhaseEqual to the sparse signal case, with the proposed convex formulation termed as compressed PhaseEqual. Different from existing convex compressed phase retrieval methods, the proposed compressed PhaseEqual formulation does not involve any regularization parameter and thus is free of the parameter tuning issue which is always tricky in practice. Order-wise recovery conditions for PhaseEqual and its sparse version (i.e. compressed PhaseEqual) are analyzed. Our theoretical results show that PhaseEqual (resp. compressed PhaseEqual) achieves perfect recovery with $\mathcal {O}(n)$ (resp. $\mathcal {O}(k\text{log}\frac{n}{k})$ ) magnitude measurements, provided that a well-correlated reference vector is available, where $k$ and $n$ denote the number of nonzero entries in the complex sparse signal and the dimension of the signal, respectively. Simulation results are provided to illustrate the effectiveness of the proposed methods.

A distributed approach to the Optimal Power Flow problem for unbalanced and mesh networks

In the present paper we introduce a new distributed optimization approach to the solution of general Optimal Power Flow problem for unbalanced and mesh networks. A new convex formulation, based on McCormick Envelopes, is proposed with a decomposition profile and a distributed approach based on proximal coordination. The resulting algorithm is shown to converge with a rate of o(1/τ), where τ is the number of iterations. The approach is validated on a modified IEEE 13 bus network, with added distributed energy resources including distributed generation and demand response.

Image Restoration by Combined Order Regularization with Optimal Spatial Adaptation.

Total Variation (TV) and related extensions have been popular in image restoration due to their robust performance and wide applicability. While the original formulation is still relevant after two decades of extensive research, its extensions that combine derivatives of first and second orders are now being explored for better performance, with examples being Combined Order TV (COTV) and Total Generalized Variation (TGV). As an improvement over such multi-order convex formulations, we propose a novel non-convex regularization functional which adaptively combines Hessian-Schatten (HS) norm and first order TV (TV1) functionals with spatially varying weight. This adaptive weight itself is controlled by another regularization term; the total cost becomes the sum of this adaptively weighted HS-TV1 term, the regularization term for the adaptive weight, and the data-fitting term. The reconstruction is obtained by jointly minimizing w.r.t. the required image and the adaptive weight. We construct a block coordinate descent method for this minimization with proof of convergence, which alternates between minimization w.r.t. the required image and the adaptive weights. We derive exact computational formula for minimization w.r.t. the adaptive weight, and construct an ADMM algorithm for minimization w.r.t. to the required image. We compare the proposed method with existing regularization methods, and a recently proposed Deep GAN method using image recovery examples including MRI reconstruction and microscopy deconvolution.

Convexifying Sparse Interpolation With Infinitely Wide Neural Networks: An Atomic Norm Approach

This work examines the problem of exact data interpolation via sparse (neuron count), infinitely wide, single hidden layer neural networks with leaky rectified linear unit activations. Using the atomic norm framework of [Chandrasekaran et al., 2012], we derive simple characterizations of the convex hulls of the corresponding atomic sets for this problem under several different constraints on the weights and biases of the network, thus obtaining equivalent convex formulations for these problems. A modest extension of our proposed framework to a binary classification problem is also presented. We explore the efficacy of the resulting formulations experimentally, and compare with networks trained via gradient descent.

Analysis and synthesis conditions for T-S fuzzy continuous-time systems with partially matched premises

Abstract A recognized challenge in the Takagi-Sugeno (T-S) fuzzy controller design concerns the use of different membership functions (MF) for controller and system. Most of the conditions available in the literature require that the controller’s MF matche the system’s one. Therefore, the implementation of such controllers may lead to unsafe operational conditions whenever such a match is lost. The main contribution of this paper is to provide new convex formulations for both stability analysis and controller design for T-S fuzzy systems under unmatched MF. We assume the same number of premises, yielding conditions called partially matched premises. To reduce conservatism, we use the Lyapunov approach, and we write the MF of the controller from the MF of the system. Two examples serve to compare our approach with others found in the literature. The achieved results suggest that our method outperforms the others.

Piecewise Linear Function Fitting via Mixed-Integer Linear Programming

Piecewise linear (PWL) functions are used in a variety of applications. Computing such continuous PWL functions, however, is a challenging task. Software packages and the literature on PWL function fitting are dominated by heuristic methods. This is true for both fitting discrete data points and continuous univariate functions. The only exact methods rely on nonconvex model formulations. Exact methods compute continuous PWL function for a fixed number of breakpoints minimizing some distance function between the original function and the PWL function. An optimal PWL function can only be computed if the breakpoints are allowed to be placed freely and are not fixed to a set of candidate breakpoints. In this paper, we propose the first convex model for optimal continuous univariate PWL function fitting. Dependent on the metrics chosen, the resulting formulations are either mixed-integer linear programming or mixed-integer quadratic programming problems. These models yield optimal continuous PWL functions for a set of discrete data. On the basis of these convex formulations, we further develop an exact algorithm to fit continuous univariate functions. Computational results for benchmark instances from the literature demonstrate the superiority of the proposed convex models compared with state-of-the-art nonconvex models.

Constrained Inverse Optimal Control With Application to a Human Manipulation Task

This paper presents an inverse optimal control methodology and its application to training a predictive model of human motor control from a manipulation task. It introduces a convex formulation for learning both objective function and constraints of an infinite-horizon constrained optimal control problem with nonlinear system dynamics. The inverse approach utilizes Bellman's principle of optimality to formulate the infinite-horizon optimal control problem as a shortest path problem and Lagrange multipliers to identify constraints. We highlight the key benefit of using the shortest path formulation, i.e., the possibility of training the predictive model with short and selected trajectory segments. The method is applied to training a predictive model of movements of a human subject from a manipulation task. The study indicates that individual human movements can be predicted with low error using an infinite-horizon optimal control problem with constraints on shoulder movement.

Space Situational Awareness Sensor Tasking: Comparison of Machine Learning with Classical Optimization Methods

The object population in the space around the Earth is subject to increase. With the advancements in sensor capabilities, it can be expected that, at the same time, more of those objects will be detected. Although this allows for significant advances in understanding of the detectable objects and the expansion of object catalogs, it also leads to significant stress on the sensor systems and makes efficient sensor tasking a prime challenge. To solve sensor tasking as an optimization problem of observing objects when a priori information is available, various methods exist. Classical methods rely on the problem being formulated in a convex representation. Computationally intensive methods based on artificial intelligence, such as machine learning, have recently gained a lot of attention and are suitable for problems even when no convex formulation can be found. In this paper, performances of a simple traditional greedy algorithm and the more complex Weapon-Target Assignment algorithm are compared with the performance of two machine learning algorithms: ant colony and distributed Q-learning. Ant colony optimization is a swarm optimization path finding methodology based on probabilistic principles; distributed Q-learning aims to find an optimal policy by maximizing the expected reward received. As an application case the observation of known objects in the geosynchronous region with a ground-based sensor is used, and performance is evaluated in terms of the number of objects successfully tracked, the computational efficiency of running the algorithms, and the difficulty of tuning the algorithms. The ant colony solutions track the most objects, whereas the greedy algorithm is the most efficient; additionally, the ant colony and distributed Q-learning require significant tuning of the algorithms before employment.