Convex Optimization Program Research Articles

Heuristics for the stochastic economic lot sizing problem with remanufacturing under backordering costs

We consider a production system where demand can be met by manufacturing new products and remanufacturing returned products, and address the economic lot sizing problem therein. The system faces stochastic and time-varying demands and returns over a finite planning horizon. The problem is to match supply with demand, while minimizing the total expected cost which is comprised of fixed production costs and inventory (holding and backordering) costs. We introduce heuristic policies for this problem which offer different levels of flexibility with respect to production decisions. We present computational methods for these policies based on convex optimization and certainty equivalent mixed integer programming, and numerically assess their cost performance and computational efficiency by means of simulation.

An Optimization and Analysis Framework for TCO Minimization of Plug-in Hybrid Heavy-duty Electric Vehicles

This paper develops an optimization framework to minimize the Total Cost of Ownership (TCO) for Plug-in Hybrid Electric Vehicles (PHEVs). In this paper, TCO is the summation of operational and main vehicle powertrain components cost. The developed optimization framework is formulated via combining convex optimization and Dynamic Programming technique. This framework aims at minimizing TCO by optimizing not only the sizing of the main powertrain components but also the powertrain topology. Using the developed optimization framework, this paper elaborates relevant design factors for a considered bus application namely: i) the value of equipping a HEV with plug-in functionality; ii) the effect of battery aging and replacement cost; iii) the sensitivity to fuel and electricity cost; Simulation results show that the TCO can be reduced by having plug-in functionality in the HEVs. However, this may not hold if the electricity price (in Euros/kWh) is higher than certain times of the fuel price (in Euros/kWh), e.g. 2.25 for the simulated cases in this paper. Simulation results also suggest that it is more profitable to equip the vehicle with a big enough battery to avoid replacing it during the vehicle economical life.

Optimization-based image reconstruction from sparsely sampled data in electron paramagnetic resonance imaging

Electron paramagnetic resonance imaging (EPRI) can yield information about the 3-dimensional (3D) spatial distribution of the unpaired-electron-spin density from which the spatial distribution of oxygen concentration within tumor tissue, referred to as the oxygen image or electron paramagnetic resonance (EPR) image in this work, can be derived. Existing algorithms for reconstruction of EPR images often require data collected at a large number of densely sampled projection views, resulting in a prolonged data-acquisition time and consequently numerous practical challenges especially to in vivo animal EPRI. Therefore, a strong interest exists in shortening data-acquisition time through reducing the number of data samples collected in EPRI, and one approach is to acquire data at a reduced number of sparsely distributed projection views from which existing algorithms may reconstruct images with prominent artifacts. In this work, we investigate and develop an optimization-based technique for image reconstruction from data collected at sparsely sampled projection views for reducing scanning time in EPRI. Specifically, we design a convex optimization program in which the EPR image of interest is formulated as a solution and then tailor the Chambolle-Pock (CP) primal-dual algorithm to reconstruct the image by solving the convex optimization program. Using computer-simulated EPRI data from numerical phantoms and real EPRI data collected from physical phantoms, we perform studies on the verification and characterization of the optimization-based technique for EPR image reconstruction. Results of the studies suggest that the technique may yield accurate EPR images from data collected at sparsely distributed projection views, thus potentially enabling fast EPRI with reduced acquisition time.

Behavioral mean-variance portfolio selection

In this paper, a behavioral mean-variance portfolio selection problem in continuous time is formulated and studied. Unlike in the standard mean-variance portfolio selection problem, the cumulative distribution function of the cash flow is distorted by the probability distortion function used in the behavioral mean-variance portfolio selection problem. With the presence of distortion functions, the convexity of the optimization problem is ruined, and the problem is no longer a conventional linear-quadratic (LQ) problem, and we cannot apply conventional optimization tools like convex optimization and dynamic programming. To address this challenge, we propose and demonstrate a solution scheme by taking the quantile function of the terminal cash flow as the decision variable, and then replace the corresponding optimal terminal cash flow with the optimal quantile function. This allows the efficient frontier and the efficient strategy to be exploited.

Energy Scheduling for Optical Channels With Energy Harvesting Devices

Optical communication systems (with visible light communication as an example) have emerged as a promising candidate for supporting high throughput requirements. Such high data rates can be achieved in a sustainable manner when energy harvesting devices are adopted for the optical transmitters. However, the existing research lacks a thorough investigation on the maximum achievable capacity for optical networks under dynamic energy conditions. This is especially true when photon counter detectors are implemented at the receiver side, and hence, optical Poisson channels are considered. To address this limitation, we develop in this paper optimal energy scheduling algorithms for optical Poisson channels with energy harvesting devices. The objective is to maximize the channel sum-rate, assuming that the side information of energy harvesting states for ${K}$ time slots is known a priori , and the battery capacity and the maximum energy consumption in each time slot are bounded. The problem is formulated as a convex optimization program with $ {{\mathcal{ O}}(K)}$ constraints, making it hard to solve using a general convex solver since the computational complexity of a generic convex solver is exponential in the number of constraints. This paper proposes an efficient energy scheduling algorithm that has a reduced computational complexity of ${{\mathcal {O}}(K^{2})}$ . The proposed algorithm is also proven to be optimal. The developed energy schedule is piece-wise constant, which changes when the battery overflows or depletes. Numerical results depict significant improvement of the optimal strategy over benchmark strategies.

Stability analysis of the linear discrete teleoperation systems with stochastic sampling and data dropout

This paper addresses the stability conditions of the sampled-data teleoperation systems consisting continuous-time master, slave, operator, and environment with discrete-time controllers over general communication networks. The output signals of the slave and master robots are quantized with stochastic sampling periods which are modeled as being from a finite set. By applying an input-delay method, the probabilistic sampling system is converted into a continuous-time system including stochastic parameters in the system matrices. The main contribution of this paper is the derivation of the less conservative stability conditions for linear discrete teleoperation systems taking into account the challenges such as the stochastic sampling rate, constant time delay and the possibility of data packet dropout. The numbers of dropouts are driven by a finite state Markov chain. First, the problem of finding a lower bound on the maximum sampling period that preserves the stability is formulated. This problem is constructed as a convex optimization program in terms of linear matrix inequalities (LMI). Next, Lyapunov–Krasovskii based approaches are applied to propose sufficient conditions for stochastic and exponential stability of closed-loop sampled-data bilateral teleoperation system. The proposed criterion notifies the effect of sampling time on the stability-transparency trade-off and imposes bounds on the sampling time, control gains and the damping of robots. Neglecting this study undermines both the stability and transparency of teleoperation systems. Numerical simulation results are used to verify the proposed stability criteria and illustrate the effectiveness of the sampling architecture.

Comparisons of Energy Management Methods for a Parallel Plug-In Hybrid Electric Vehicle between the Convex Optimization and Dynamic Programming

This paper proposes a comparison study of energy management methods for a parallel plug-in hybrid electric vehicle (PHEV). Based on detailed analysis of the vehicle driveline, quadratic convex functions are presented to describe the nonlinear relationship between engine fuel-rate and battery charging power at different vehicle speed and driveline power demand. The engine-on power threshold is estimated by the simulated annealing (SA) algorithm, and the battery power command is achieved by convex optimization with target of improving fuel economy, compared with the dynamic programming (DP) based method and the charging depleting–charging sustaining (CD/CS) method. In addition, the proposed control methods are discussed at different initial battery state of charge (SOC) values to extend the application. Simulation results validate that the proposed strategy based on convex optimization can save the fuel consumption and reduce the computation burden obviously.

Optimization-based synthesis of stochastic biocircuits with statistical specifications.

Model-guided design has become a standard approach to engineering biomolecular circuits in synthetic biology. However, the stochastic nature of biomolecular reactions is often overlooked in the design process. As a result, cell-cell heterogeneity causes unexpected deviation of biocircuit behaviours from model predictions and requires additional iterations of design-build-test cycles. To enhance the design process of stochastic biocircuits, this paper presents a computational framework to systematically specify the level of intrinsic noise using well-defined metrics of statistics and design highly heterogeneous biocircuits based on the specifications. Specifically, we use descriptive statistics of population distributions as an intuitive specification language of stochastic biocircuits and develop an optimization-based computational tool that explores parameter configurations satisfying design requirements. Sensitivity analysis methods are also performed to ensure the robustness of a biocircuit design against extrinsic perturbations. These design tools are formulated with convex optimization programs to enable rigorous and efficient quantification of the statistics. We demonstrate these features by designing a stochastic negative feedback biocircuit that satisfies multiple statistical constraints and perform an in-depth study of noise propagation and regulation in negative feedback pathways.

Privacy-preserving MaaS fleet management

On-demand traffic fleet optimization requires operating Mobility as a Service (MaaS) companies such as Uber, Lyft to locally match the offer of available vehicles with their expected number of requests referred to as demand (as well as to take into account other constraints such as driver’s schedules and preferences). In the present article, we show that this problem can be encoded into a Constrained Integer Quadratic Program (CIQP) with block independent constraints that can then be relaxed in the form of a convex optimization program. We leverage this particular structure to yield a scalable distributed optimization algorithm corresponding to computing a gradient ascent in a dual space. This new framework does not require the drivers to share their availabilities with the operating company (as opposed to standard practice in today’s mobility as a service companies). The resulting parallel algorithm can run on a distributed smartphone based platform.

A Truthful Mechanism for the Generalized Assignment Problem

We propose a truthful-in-expectation, (1-1/ e )-approximation mechanism for a strategic variant of the generalized assignment problem (GAP). In GAP, a set of items has to be optimally assigned to a set of bins without exceeding the capacity of any singular bin. In the strategic variant of the problem we study, values for assigning items to bins are the private information of bidders and the mechanism should provide bidders with incentives to truthfully report their values. The approximation ratio of the mechanism is a significant improvement over the approximation ratio of the existing truthful mechanism for GAP. The proposed mechanism comprises a novel convex optimization program as the allocation rule as well as an appropriate payment rule. To implement the convex program in polynomial time, we propose a fractional local search algorithm which approximates the optimal solution within an arbitrarily small error leading to an approximately truthful-in-expectation mechanism. The proposed algorithm improves upon the existing optimization algorithms for GAP in terms of simplicity and runtime while the approximation ratio closely matches the best approximation ratio known for GAP when all inputs are publicly known.

Complex performance control using sliding mode fuzzy approach for discrete-time nonlinear systems via T-S fuzzy model with bilinear consequent part

In this paper, a stabilization problem for the discrete nonlinear system with external disturbance, multiplicative noises and multiple constraints has been discussed in accordance with the definition of Lyapunov stability. Based on fuzzy modeling approach, the overall fuzzy model of a nonlinear plant is transformed into a class of linear systems. Applying a Sliding Mode Fuzzy Control (SMFC) scheme, the designed controller causes the closed-loop system converging to the sliding surface and achieving the required control performance. For the control performance, the concepts of stability, individual state variance and passivity constraints are introduced for the sliding mode fuzzy control system. To apply convex optimal programming algorithm, some sufficient conditions derived in this paper are reduced to Linear Matrix Inequality (LMI) problem. At last, two simulation examples are proposed to demonstrate the applicability and usefulness of the proposed design method. One of the examples is to discuss the conservatism of this paper. Another is to show that the discrete truck-trailer system controlled by sliding mode fuzzy controller can achieve stability constraints, individual state variance constraints and passivity constraints.

Key-frame Extraction With Semantic Graphs in Assembly Processes

In this letter, we present a novel method for key-frame extraction in assembly processes, with the support of visual sensors. The method is based on the construction of a sequence of weighted graphs containing the semantics of the underlying scene, encoded as primitive temporal and spatial relations between the involved objects. The objects are modeled as three-dimensional ellipsoids, with the formulation of the minimum volume covering ellipsoid problem as a convex optimization program, which is solved by an efficient algorithm. The new models are used in the sequence of actions–relations to increase the accuracy of the key-frame extraction, by extending the states encoded in the graphs. We present and analyze the method, followed by preliminary experimental results, and provide significant evidence on its potential.

Market-Driven Energy Storage Planning for Microgrids with Renewable Energy Systems Using Stochastic Programming

Battery Energy Storage Systems (BESS) can mitigate effects of intermittent energy production from renewable energy sources and play a critical role in peak shaving and demand charge management. To optimally size the BESS from an economic perspective, the tradeoff between BESS investment costs, lifetime, and revenue from utility bill savings along with microgrid ancillary services must be taken into account. The optimal size of a BESS is solved via a stochastic optimization problem considering wholesale market pricing. A stochastic model is used to schedule arbitrage services for energy storage based on the forecasted energy market pricing while accounting for BESS cost trends, the variability of renewable energy resources, and demand prediction. The uniqueness of the approach proposed in this paper lies in the convex optimization programming framework that computes a globally optimal solution to the financial trade-off solution. The approach is illustrated by application to various realistic case studies based on pricing and demand data from the California Independent System Operator (CAISO). The case study results give insight in optimal BESS sizing from a cost perspective, based on both yearly scheduling and daily BESS operation.

Strategic Bidding for Producers in Nodal Electricity Markets: A Convex Relaxation Approach

Strategic bidding problems in electricity markets are widely studied in power systems, often by formulating complex bi-level optimization problems that are hard to solve. The state-of-the-art approach to solve such problems is to reformulate them as mixed-integer linear programs (MILPs). However, the computational time of such MILP reformulations grows dramatically, once the network size increases, scheduling horizon increases, or randomness is taken into consideration. In this paper, we take a fundamentally different approach and propose effective and customized convex programming tools to solve the strategic bidding problem for producers in nodal electricity markets. Our approach is inspired by the Schmudgen's Positivstellensatz Theorem in semialgebraic geometry; but then we go through several steps based upon both convex optimization and mixed-integer programming that results in obtaining close to optimal bidding solutions, as evidenced by several numerical case studies, besides having a huge advantage on reducing computation time. While the computation time of the state-of-the-art MILP approach grows exponentially when we increase the scheduling horizon or the number of random scenarios, the computation time of our approach increases rather linearly.

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...

Privacy-preserving MaaS fleet management

Abstract On-demand traffic fleet optimization requires operating Mobility as a Service (MaaS) companies such as Uber, Lyft to locally match the offer of available vehicles with their expected number of requests referred to as demand (as well as to take into account other constraints such as driver’s schedules and preferences). In the present article, we show that this problem can be encoded into a Constrained Integer Quadratic Program (CIQP) with block independent constraints that can then be relaxed in the form of a convex optimization program. We leverage this particular structure to yield a scalable distributed optimization algorithm corresponding to computing a gradient ascent in a dual space. This new framework does not require the drivers to share their availabilities with the operating company (as opposed to standard practice in today’s mobility as a service companies). The resulting parallel algorithm can run on a distributed smartphone based platform.

Multi-constrained fuzzy intelligent control for uncertain discrete systems with complex noises: an application to ship steering systems

ABSTRACTIn this paper, a performance constrained fuzzy controller design is investigated for the nonlinear uncertain discrete-time ship steering system with complex noises. The performance constraints considered in this paper include individual state variance constraint and passivity constraint. The nonlinear discrete-time ship steering system considered in this paper is represented by a discrete-time Takagi–Sugeno fuzzy model with perturbations, additional noise and multiplicative noise. According to this complex nonlinear discrete-time stochastic system, a fuzzy controller design approach is investigated based on Lyapunov theory, passivity theory and covariance control theory. Some sufficient conditions are derived to satisfy stability constraint, individual state variance constraint and passivity constraint, simultaneously. These sufficient conditions are constructed as linear matrix inequality forms that can be solved by the convex optimal programming algorithm. Finally, some simulation results are provided to verify the validity and applicability of the proposed fuzzy control approach with multiple performance constraints.

Optimization-based image reconstruction with artifact reduction in C-arm CBCT

We investigate an optimization-based reconstruction, with an emphasis on image-artifact reduction, from data collected in C-arm cone-beam computed tomography (CBCT) employed in image-guided interventional procedures. In the study, an image to be reconstructed is formulated as a solution to a convex optimization program in which a weighted data divergence is minimized subject to a constraint on the image total variation (TV); a data-derivative fidelity is introduced in the program specifically for effectively suppressing dominant, low-frequency data artifact caused by, e.g. data truncation; and the Chambolle–Pock (CP) algorithm is tailored to reconstruct an image through solving the program. Like any other reconstructions, the optimization-based reconstruction considered depends upon numerous parameters. We elucidate the parameters, illustrate their determination, and demonstrate their impact on the reconstruction. The optimization-based reconstruction, when applied to data collected from swine and patient subjects, yields images with visibly reduced artifacts in contrast to the reference reconstruction, and it also appears to exhibit a high degree of robustness against distinctively different anatomies of imaged subjects and scanning conditions of clinical significance. Knowledge and insights gained in the study may be exploited for aiding in the design of practical reconstructions of truly clinical-application utility.

Portfolio Construction Based on Stochastic Dominance and Empirical Likelihood

We develop and apply a portfolio optimization framework based on the Stochastic Dominance (SD) decision criterion and the Empirical Likelihood (EL) estimation method. SD and EL share a distribution-free assumption structure that allows for dynamic and non-Gaussian return distributions. The combination of SD and EL can be implemented with standard convex optimization and Linear Programming methods. We apply the SD/EL framework to a range of equity industry momentum strategies. Our moment conditions are based on stylized facts about common risk factors in the stock market. SD/EL yields important ex-ante performance improvements relative to heuristic diversification and optimization based on the Mean-Variance criterion and/or plug-in estimates. Relative to the CRSP all-share index, SD/EL improves average out-of-sample return by more than eight percentage points per annum, with less downside risk, semi-annual rebalancing and no short sales.

Stability of discrete-time switching systems with constrained switching sequences

We introduce a novel framework for the stability analysis of discrete-time linear switching systems with switching sequences constrained by an automaton. The key element of the framework is the algebraic concept of multinorm, which associates a different norm per node of the automaton, and allows to exactly characterize stability. Building upon this tool, we develop the first arbitrarily accurate approximation schemes for estimating the constrained joint spectral radius ρˆ, that is the exponential growth rate of a switching system with constrained switching sequences. More precisely, given a relative accuracy r>0, the algorithms compute an estimate of ρˆ within the range [ρˆ,(1+r)ρˆ]. These algorithms amount to solve a well defined convex optimization program with known time-complexity, and whose size depends on the desired relative accuracy r>0.