Optimal Assigner Decisions in a Hybrid Predictive Control of an Autonomous Vehicle in Public Traffic

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1 016 American Control Conference (ACC) Boton Marriott Coley Place July 6-8, 016. Boton, MA, USA Otimal Aigner Deciion in a Hybrid Predictive Control of an Autonomou Vehicle in Public raffic Qian Wang 1, Student Member, IEEE, Behah Ayalew 1, Member IEEE and homa Weikircher Abtract he comlex ublic traffic environment reuire an autonomou vehicle to have the ability of lanning and executing a euence of different maneuver, uch a maintaining cruiing eed, changing eed to follow a vehicle in front or to lead a vehicle in the rear, and changing a lane when neceary and oible. hi aer reent a hierarchical hybrid redictive control framework that integrate the dicrete otimization roblem of maneuver election with article motion-baed model redictive trajectory guidance for an autonomou road vehicle. o addre the challenge of olving the reulting mixed integer nonlinear rogramming (MINP) roblem efficiently for online imlementation, a relaxation method i introduced that tranform the MINP roblem into a nonlinear rogramming roblem with imroved feaibility for online imlementation. he erformance of the rooed framework i illutrated via imulation of the autonomou vehicle in highway cenario. I. INRODUCION he baic tak facing the controller of an autonomou vehicle in ublic traffic i the lanning and guidance of the vehicle motion while raidly and ytematically accommodating a number of oibly changing contraint. hee include tire/road friction limit, vehicle actuation limit, avoiding tationary and moving obtacle, obeying traffic rule, conidering aenger comfort, and o on. hi tak mut be addreed by deigning comutationally efficient motion lanning algorithm that can determine the bet euence of the vehicle maneuver and contituent motion while atifying the lited contraint. Model redictive control (MPC)-baed motion lanning i receiving ignificant attention becaue it able to generate feaible trajectorie by olving control and tate contrained otimization in a receding rediction horizon. In [1], a highfidelity nonlinear vehicle model wa imlemented in a nonlinear MPC (or NMPC) for motion lanning, but the comutational burden wa found to be too high in the reence of tightened contraint uch a needed for obtacle avoidance. o reduce the comutational comlexity, a linear time-variant (LV) model wa ued in [] to aroximate the vehicle dynamic for on-line MPC. hi reulted in a multiarametric rogramming roblem. Previou work in [3] rooed a hierarchical NMPC framework which decomoed the motion lanning work into a high-level MPC-baed re-lanner that ue a low-fidelity oint-ma vehicle model and a low-level MPC-baed follower that ue the high-fidelity vehicle model emloyed in [1, ]. In addition to oible hierarchical model conitency iue, 1 Qian Wang and Behah Ayalew are with the Alied Dynamic& Control Grou at the Clemon Univerity International Center for Automotive Reearch (CU-ICAR), 4 Reearch Dr., 9607, Greenville, SC,USA, {wang8, behah}@clemon.edu homa Weikircher i currently at Daimler R&D, Germany. the comutational burden of the two MPC otimization loo of thi aroach could be rohibitive for ractical ue. In [4], a reduced-order nonlinear article motion model i ued to deign a high-level NMPC and either the redicted control outut or the redicted tate trajectorie were configured to be tracked uing traditional vehicle dynamic controller at the lower-level [5]. hi aroach howed good erformance in ath tracking and obtacle avoidance, but the olver could get traed at local minima in comlex traffic environment if the etu for the otimal control roblem in the NMPC i not re-configured by a higher-level deciion making module. o thi end, in [6], the author introduced a hierarchical hybrid ytem control framework with a et of finite tate machine (FSM) embedded in the higher-level deciionmaking module (called aigner module) above the NMPC. herein, the aigner module ue the information from a navigation module (route), urrounding traffic ign or obtacle and other contraint to elect dicrete maneuver tate through re-defined witching rule. In the reent aer, we configure thee deciion of the aigner module to be elected otimally and redictively o a to change the maneuver tate and the etu of the NMPC a freuently a neceary for the revailing driving cenario. Similar hierarchical deign can be found in lenty of reviou work. A hierarchical FSM concet with meta-tate and ub-tate machine for vehicle maneuver tate wa deigned in [7] and ha been imlemented in an autonomou vehicle in the DARPA Urban Challenge [8]. In [9], a rulebaed automaton (FSM) wa deigned to regulate the longitudinal motion maneuver of autonomouly controlled vehicle (ACV) to avoid colliion under cruiing and merging cenario. A game theory method wa ued in [10] to deign a robut hybrid controller, which guarantee afety in vehicle latooning under ome uncertaintie. he method wa alied later by [11] with non-determinitic automaton to regulate an interection roblem. In mot of thee exiting work, the dicrete deciion of maneuver election were not done otimally in a redictive horizon fahion. he otimization of uch hybrid ytem involve olving for the otimal dicrete tate (maneuver) a well a their otimal local-control inut (uch a acceleration) [1]. hi lead to a mixed-integer nonlinear rogramming roblem (MINP) [13], which reuire ecific heuriticbaed olver to find a olution. However, the NP-comlete nature of the MINP roblem make it hard to olve efficiently for real-time imlementation, eecially when the roblem at hand i comlex with nonlinearity and fat dynamic [14]. hi i the cae for ACV in many ublic traffic cenario. Another contribution of the reent aer, therefore, i the introduction of relaxation contraint to tranform the MINP into a regular nonlinear rogramming, /$ AACC 3468

2 which can be olved efficiently in milliecond time-cale with multi-hooting method and auto-generated code uitable for real-time imlementation [15]. he ret of the aer i organized a follow. Section II introduce the rooed hierarchical hybrid redictive control framework. Section III introduce reformulation of the MPC roblem for maneuver lanning, dicue how it reult in a MINP roblem and outline the relaxation method ued to aroximate and olve it. Reult and dicuion are included in Section V to illutrate the working of the rooed framework and brief concluion are offered in Section VI. II. HYBRID PREDICIVE CONROL FRAMEWORK A. Hierarchical Framework Fig. 1 how a chematic of the rooed hierarchical hybrid control framework for the autonomou road vehicle. It conit of the route navigator module, the environment recognition module, the higher-level maneuver lanning module and the lower-level maneuver execution module. Figure. 1. Hierachical control framework he route navigator module (uch a a GPS navigator) lan the route from initial oition to the final oition baed on a ma and localization of the controlled vehicle. he environment information, uch a lane mark, traffic ign or ignal, the ize or tate of moving object, are aumed catured in the environment recognition module via camera, radar, lidar or wirele device (VV, VI). he detail of thee two module are beyond the coe of thi aer and their outut information i imly aumed available to the high-level maneuver lanner. he aigner module i reonible for the election of the dicrete maneuver tate. Different maneuver are deigned and groued a (et of) finite tate machine () for the aigner to elect from. At each time te, the aigner can chooe a maneuver for each lane uing the information from the navigation and environmental module. he deciion on the maneuver for lane election and the generation of the neceary trajectorie and/or control outut (e.g. acceleration) for the elected maneuver i then comuted by the MPC-baed redictive trajectory guidance (PG) ytem. he MPC in the PG ytem ue a D curvilinear article motion decrition of the vehicle, the aociated ath reference, obtacle decrition, tire-road friction contraint and/or traffic rule (ee, e.g.[4]). From the aigner, we aociate ecific objective function etu (maneuver-tracking reference and correonding weight) with each maneuver and a them to a ingle veratile MPC configuration of the PG. In our reviou work [6], a rulebaed maneuver election trategy ha been ued inide the aigner, which elected a ingle maneuver and merely aed the correonding objective function etu to the PG. In thi aer, we aly rule to re-elect a et of maneuver candidate for each lane, and then we introduce otimality for further election of the bet maneuver from thi et. o thi end, we reformulate the MPC roblem in the PG with a new objective function et u. hi endow the PG with the ability to intelligently witch the maneuver lan within the rediction horizon. Effectively, thi arrangement move art of the maneuver election tak of the aigner to the PG (Fig.1). It alo convert the otimization roblem for the MPC into a MINP roblem. We hall detail thi in Section III. he control outut comuted by the PG will be ent to the lower-level controller of the continuou vehicle dynamic for execution via the available lower-level vehicle dynamic controller, whoe dicuion i omitted here. he reader i referred to [5] and other tandard reference for thi toic. B. Hybrid Sytem Modeling of the Vehicle Maneuver Here, we briefly dicu the modeling of the vehicle motion maneuver a a hybrid ytem. A hybrid ytem involve both continuou dynamic and dicrete tate dynamic whoe evolution deend on each other. It can often be viewed a a collection Σ of indexed continuou dynamical ytem. here often exit ome witching or juming ma among them, uch a when the tate enter a ecified ubet of the tate ace, named a jum et. he jum et can be defined either by the inherent evolution of the ytem (autonomou jum et) or by external control command (controlled jum et) [1]. In thi vein, the autonomou road vehicle witching or juming between different maneuver can be characterized a a hybrid ytem with the automaton deicted in Fig.. In general, the aigner module may conit of everal finite tate machine (FSM) at the to. Baed on the cenario the ACV i in, a relevant FSM will be choen. In each FSM, for examle for the highway cae hown, the ACV can jum or witch it maneuver among the contituent dynamical ytem denoted by the ellitical node (ee maneuver definition in aragrah below). Each arrowed-edge rereent a jum between the different maneuver with the label decribing the reuired juming ma/condition. he black dahed edge rereent jum defined by re-defined embedded witching rule like thoe lited in our reviou work [6], while the olid red edge indicate otimizationbaed jum we detail in the reent aer. Both of thee et of jum can be viewed a autonomou jum. 3469

3 integrating ome of the aigner function. he vehicle tate and reference ath defined in the Frenet frame are hown in Fig. 4. he motion of the article/vehicle with reect to the local reference ath (lane centerline) i given by the angular alignment error and lateral error. he following euation ummarize the reulting nonlinear dynamic model decribing the motion a well a the evolution of the ath coordinate : v a t t Figure.. Maneuver automaton in the aigner. Rule-baed jum et are denoted by R and Otimization-baed one by O. For the highway FSM examle, for each lane, we define three reference maneuver (ee the blue ellitical node in Fig.) for the ACV: normal tracking, following or leading. he jum et between thee maneuver are defined by target eed aignment rule hown in Fig 3. hee jum are denoted by R in Fig.. A normal tracking maneuver i defined a tracking a normal cruie eed v ref. A following maneuver for the ACV refer to tracking the eed v t,of of a detected front object vehicle on the ame lane with v t,of < v ref, and a leading maneuver refer to tracking the eed v t,or of a detected rear vehicle on the ame lane with v t,or > v ref. hee are deicted in Fig. 3, where v t i the longitudinal eed of the ACV, v t,r i the reference/target eed of the ACV aigned to the ecific lane. A lane change maneuver i defined by witching the reference lane from the current one to another. he jum tranition to thi maneuver will be determined by the otimization in the (re-formulated MPC of the) PG module baed on the target eed for each lane and the current tate of the ACV. v co e t e 1 y y v in a a a /, v () / t, d 1 v co t e 1 y e In thee motion euation, the deired acceleration a t,d and the deired deviation from the reference yaw rate are the treated a the inut ued to control the d, article along the ath. he reference ath curvature κ i aumed to be known along the reference ath coordinate. v t, a t are the article eed and acceleration along the ath. i the yaw rate, ψ e, y e are the alignment error and lateral error to the reference ath, and a t, are the time contant of the firt-order aroximation of the longitudinal and lateral vehicle dynamic. i the rojection of the eed v t on the reference ath. e t e t t d t a t e Figure 4. Particle motion decrition for the vehicle he nearby object (obtacle or moving vehicle) are alo conidered on the road, a hown in Fig. 5. he motion of object i i given by: Figure 3. arget eed aignment. he witching of target eed v t,r for each lane can alo be conidered a a controlled jum when it i imoed by traffic ign and other exogenou traffic control device. hi cae i not urued in thi aer. III. MPC PROBLEM RE-FORMULAION A. Particle Motion Baed Vehicle Model he D curvilinear article motion baed MPC rooed in [4] i re-formulated here a maneuver lanner v t a t oi oi,0 t, oi t, oi 1 y y v t a t e, oi e, oi,0 n, oi n, oi 1 where, t i the internal time in the rediction model (of the MPC). Euation (7) and (8) are then ued to etimate the oition of the object in the rediction horizon baed on the current meaurement of the longitudinal velocity v, longitudinal acceleration to, i to, i a, lateral velocity v and no, i 3470

4 lateral acceleration no, i a in ath coordinate. he a and no, i to, i a are held contant for the rediction horizon, but are to be udated at each MPC udate. he initial oition of object i (at rediction) are denoted ( o,0, y e, o,0 ). B. Contraint he contraint to kee a afe ditance between the ACV and any nearby object i i modeled by the ellitic ineuality: y y e e, oi oi y f e, oi o,, Do Do i i 1 which i alo deicted in Fig. 5. ζ Do 0 i a lack variable, which allow the olver to find a feaible olution in emergency ituation. f ζ,do i an otional tuning arameter (ha a unit of time). Δy e,oi and Δ o, are calculated by incororating the geometry (length and width) of the object and the ACV. hee are aumed available from ening and/or VV communication. he control inut mut alo be limited to hyical contraint of the acceleration according to the friction ellie of a real vehicle tire/road contact: v v, / a t t r n, gg a t, d g H gg Here, μ H i the limiting tire-road friction coefficient, g i the gravitational contant. a n,gg [0,1] i the caling of the ellie for lateral acceleration. he lack variable ζ gg enable the formulation of the limit value of the combined acceleration a a oft contraint. Other tate contraint like the lane boundarie, eed limit and the minimum turning radiu, etc are alo conidered but not decribed here due to lack of ace. For comlete detail, leae refer to [4]. C. MPC Problem Re-Formulation Unlike reviou work [4], where a fixed uite of reference are conidered in the MPC objective for the redictive trajectory guidance(pg) module, here we configure the MPC to have the ability to intelligently witch between maneuver tate and their correonding reference. o thi end, we introduce a lane index and an aociate weight variable Z. he otimization roblem to be olved over the rediction horizon [0, H ] i then re-formulated a: min Figure 5. Object motion definition in road reference frame N N N 1 Z, y r k 1. k 1,, k y r u. k, k (11) P k R xk, u k, Z P1,k k 1 Q k 1 k 0 ubject to : x f x, u, Z x(0) x0 u U x X 0 c x, u Z 1, Z 0,1 Q Here, k (0, 1,, N ), where N i the rediction te length, i the euiditantly amled time index of the dicrete time verion of the continuou ytem (1) rereenting the vehicle model given by (1)-(6) and auxiliary euation(integrator) for the lack variable. x cover all the tate variable of the article motion and lack variable; and X rereent the tate ace for x. x 0 denote the current/initial tate. he rediction horizon H i defined by H = N Δ and Δ i the minimum amle time of the MPC. i lane index and Q i the et of lane. r 1,, contain the eed and lateral oition reference for maneuver candidate for each lane and r contain reference for the lack variable. P 1, P and R are the weighting matrice for the candidate maneuver tracking error, lack variable reference tracking error and control effort, reectively. y=[y 1, y ] i the ytem outut, including the eed v t and lateral oition y e of the ACV which are groued in y 1, and the lack variable outut ζ Do, ζ gg, groued in y. See (19- ) in the next ection for an examle of how thee variable are defined. he control variable u i alied in a iecewie contant fahion, a u k, and only the firt te u 0 will be ued to control the ytem for each MPC udate te. In (1), U denote the admiible et for u. All the nonlinear contraint uch a the colliion avoidance and friction limit are included in (14). In the otimization roblem (11), Z i the weight variable for election of the -th lane and the reference eed on that lane denoted by r 1,. At each time k, only one maneuver i choen, o Z hould atify the integrality contraint (15). hi formulation lead to a MINP roblem to be olved at each MPC udate te. D. Relaxation Method he MINP roblem i hard to olve efficiently at real time ince it i NP-comlete. But ome relaxation can be made to aroximate the roblem a a nonlinear rogramming roblem that i tractable for real-time imlementation. hi i achieved by extending the feaible olution et of the roblem by relaxing the integrality contraint in (15) a: Z 1, Z [0,1] Q where, the weight variable Z become a real variable which can then be regarded a an additional tate in (1) with the auxiliary dynamic: z Z u 3471

5 where u z i a virtual inut ued to maniulate Z. hi become one art of the control vector u in (14). hen, the new otimization roblem at each MPC te i given by: min N N N 1 Z, y r k 1. k 1,, k y r u. k, k (0) P k R xk, u k 1 Q P1 k 1 k 0 k ubject to the modified contraint, including (18) and (19). A. Simulation Setting IV. RESULS AND DISCUSSIONS o illutrate the erformance of the rooed hybrid redictive control aroach, we conider two highway cenario on a ection with three traight lane, ee Fig. 6. Figure 6. Highway cenario with 3 lane he jum event of a lane change maneuver i activated baed on the value of weight variable Z otimized according to (18). For the lane in Fig. 6 and the finite et of maneuver in Fig., all the dicrete maneuver for the hybrid ytem are lited in able I. State index ABLE I. DISCREE MANEUVER SAES Maneuver Decrition 1 Normal racking in Lane 1 Following in Lane 1 3 Leading in Lane 1 4 Normal racking in Lane 5 Following in Lane 6 Leading in Lane 7 Normal racking in Lane 3 8 Following in Lane 9 Leading in Lane 3 Weight Coefficient For the three-lane highway cenario, Q={1,, 3}, Q. he reference tate of each candidate maneuver will be aigned in otimization etu of the reformulated MPC. o thi end, the ret of the term in (0) are given by: y y, v 1, k e, k t, k y Z, Z, Z,,, k 1, k, k 3, k gg, k Do, k r y, v 1,, k e, r,, k t, r,, k r 1,1,1,, v, k gg, r, k t, k u a,, u, u, u k t, d, k, d, k z1 z z 3 Due to the roblem formulation in MPC, the otimization can be traed in one lane with following maneuver or a leading maneuver intead of normal tracking. Z 1 Z Z 3 hi might lead to ignificant deviation from the deired eed v ref,. A thee maneuver rereent art of the global minimum of the roblem, change mut be made to the objective function to force the tate flow jum from the current minimum and converge to other deired global minimum. herefore, a forced maneuver from the maneuver candidate i added to the MPC by making it weight variable Z track a value of 1 in (0) and (), with related weight z in P for control activation. he jum et to thi ecial maneuver i defined a the violation of a tolerance eed range [v lcl, v lch ] around v ref with v lcl v ref v lch and the availability of an oen lane for normal tracking: z,, w, if v v, v v v 0, ele z t lcl lch t r ref, 1,, 3 where w z i a oitive value large enough to activate the forced lane change. o eliminate the redundancy of electing multile maneuver a the forced maneuver, a riority of the lane election hould be followed for lane with z =w z in decending order from lane, lane 3 to lane 1. hu, a z with lower riority will be reet to zero. Such a riority deign kee the ACV referring lane (middle lane) for cruie eed tracking. he riority can alo be et to accommodate other local traffic rule/cutom. he control aroach and imulation etting above are alied to controlling the high fidelity vehicle model ued by [8] in two highway cenario with v ref = 30m/. he nonlinear rogramming roblem in the re-formulated MPC i olved efficiently via the ACADO oolkit [16] which imlement euential uadratic rogramming. For detail, alo ee [17]. B. Reult Fig. 7 how the reult for cenario 1: the ACV with a fater initial eed at 30m/ change lane from lane 3 to lane (Z 3 reduce and Z rie u ignificantly) and follow the lower object vehicle (OV), OV, traveling with a eed of 5 m/ to avoid the OV3 in the front on lane 3 travelling at the even lower eed of 0 m/. In thi rocedure, ACV doen t change to lane 1 becaue OV1 occuie lane 1 in arallel with OV at the ame forward eed, which make following OV1 more cotly. he obervation of the ACV lowing down to the eed, which violate the tolerance eed range [7.5, 3.5] m/ around v ref i due to the unavailability of all lane. When it ae OV3 and lane 3 become available for normal tracking, a forced maneuver of changing the lane back to lane 3 and accelerating to track v ref i activated. Finally, the ACV ae OV via lane 3. Fig. 8 how the highway cenario to avoid a fater rear OV3 that change lane uddenly. Initially, ACV goe on lane 3 at eed of 30m/ in arallel with OV on lane. When the ACV detect OV3 aroaching at 35m/ from behind. he reference eed for lane 1 i witched to 35m/ and the controller decide to change lane to lane to minimize the global cot. However, a lane ha been occuied by OV nearby, the ACV ha to accelerate and bya the ellitical colliion avoidance boundary (9) to 347

6 reach lane. During the bya, the ACV goe a long way to change lane from lane 3 to lane, thu Z change lowly from 0 to 1. At around t=6, when the ACV uccefully reache lane and OV3 catch u with the ACV in longitudinal direction, OV3 uddenly change it lane from lane 3 to lane 1, which force the ACV to brake harly to avoid colliion. Figure 7. Highway cenario 1 Figure 8. Highway cenario Note that in both cenario, the MPC execution time are of the order of 40m or le; increaing mainly when the obtacle ineuality contraint are engaged. V. CONCLUSION AND FUURE WORK hi aer reented a hierarchical hybrid redictive control framework that integrate the dicrete otimization roblem of maneuver election with article motion-baed model redictive trajectory guidance for an autonomou road vehicle. A convex relaxation method i ued to tranform the reulting MINP roblem into a nonlinear rogramming roblem with imroved feaibility for online imlementation. wo highway cenario imulation of the ACV controlled with thi framework have been included to how how it uccefully guide the vehicle with reaonable MPC execution time. Future work will focu on: 1) Extending the otimality to the re-election of the maneuver candidate; ) More trategie for eliminating local minimum in the domain of the objective function; 3) Stability, robutne and convergence analyi of the framework. REFERENCES [1]. F. Borrelli, P. Falcone,. Keviczky, J. Agari, and D. Hrovat. "MPCbaed aroach to active teering for autonomou vehicle ytem."international Journal of Vehicle Autonomou Sytem vol.3, no. -4, 005, []. P. Falcone, F. Borrelli, J. Agari, H. E. eng and D. Hrovat. Predictive active teering control for autonomou vehicle ytem. Control Sytem echnology, IEEE ranaction on, vol.15, no.3, 007, [3]. P. Falcone, F. Borrelli, H. E. eng, J. Agari, and D. Hrovat. "A hierarchical model redictive control framework for autonomou ground vehicle." In American Control Conference, 008, IEEE, 008. [4].. Weikircher and B. Ayalew. "Predictive Control for Autonomou Driving in Dynamic Public raffic," Proceeding of the American Control Conference, July 1-3, Chicago, IL [5].. Weikircher and B. Ayalew. "Framework for Interfacing rajectory racking with Predictive rajectory Guidance for Autonomou Road Vehicle." Proceeding of the American Control Conference, July 1-3, Chicago, IL [6]. Q. Wang,. Weikircher, B. Ayalew, Hierarchical Hybrid Predictive control of an Autonomou Road Vehicle, Proceeding of the ASME 015 Dynamic Sytem and Control Conference, Oct. 8-30, Columbu, OH. [7]. K. Arda, and Ü. Özgüner. Hierarchical finite tate machine for autonomou mobile ytem. Control Engineering Practice vol. 1, no., 013, [8]. C. Urmon, et al. "artan racing: A multi-modal aroach to the DARPA Urban Challenge.",007. [9]. A. Girault. A hybrid controller for autonomou vehicle driving on automated highway. ranortation Reearch Part C: Emerging echnologie, vol. 1, no. 6,.41-45, Dec. 004 [10]. L., John, D. N. Godbole, and S. Satry. "Verified hybrid controller for automated vehicle." Automatic Control, IEEE ranaction on vol. 43, no. 4, 1998, [11]. V. Rajeev, and D. Del Vecchio. "Safety control of hidden mode hybrid ytem." Automatic Control, IEEE ranaction on, vol. 57, no. 1, 01, [1]. M. S. Branicky, V. S. Borkar, and S. K. Mitter. "A unified framework for hybrid control: Model and otimal control theory." Automatic Control, IEEE ranaction on, vol. 43, no. 1, 1998, [13]. P. Belotti, et al. Mixed-integer nonlinear otimization. Acta Numerica, no., 013, [14]. A. Richard, and J. How. "Mixed-integer rogramming for control."american Control Conference, 005. Proceeding of the 005. IEEE, 005. [15]. M. Vukov, et al. "Auto-generated algorithm for nonlinear model redictive control on long and on hort horizon." Deciion and Control (CDC), 013 IEEE 5nd Annual Conference on. IEEE, 013. [16]. B. Houka, H. J. Ferreau, and M. Diehl. "ACADO toolkit An oenource framework for automatic control and dynamic otimization."otimal Control Alication and Method, vol. 3, no. 3, 011, [17]. H. J. Ferreau, et al. "OASES: A arametric active-et algorithm for uadratic rogramming". Mathematical Programming Comutation, vol. 6, no. 4, 014,