On the Complexity of Explicit MPC Laws

Transcription

1 On the Complexity of Explicit MPC Laws Francesco Borrelli, Mato Baotić, Jaroslav Pekar and Greg Stewart Abstract Finite-time optimal control problems with quadratic performance index for linear systems with linear constraints can be transformed into Quadratic Programs (QPs). Model Predictive Control requires the online solution of such QPs. This can be obtained by using a QP solver or evaluating the associated explicit solution. Objective of this note is to shed some light on the complexity of the two approaches. I. INTRODCTION In [3], [2] the authors have shown how to compute the solution to constrained finite-time optimal control (CFTOC) problem for discrete-time linear systems as a piecewise affine (PWA) state-feedback law. Such a law is computed off-line by using a multi-parametric programg solver [3], [4], [9], which divides the state space into polyhedral regions, and for each region deteres the linear gain and offset which produces the optimal control action. This method reveals its effectiveness when a Model Predictive Control (MPC) strategy is used [0]. At each sampling time the MPC requires the solution of an open-loop CFTOC problem, which, for a quadratic performance index and known (measured) system state, corresponds to solving a Quadratic Program (QP). Having a precomputed solution as an explicit piecewise affine function of the state vector reduces the on-line computation of the MPC control law to a function evaluation, thus avoiding the on-line solution of a quadratic program. The objective of this note is to shed some light on the complexity of the on-line solution of a quadratic program (by means of an active set QP solver) versus the on-line evaluation of the explicit solution (by means of an explicit solver ). We will focus on the three main components of an active set QP algorithm and of an explicit solver: () the amount of stored data, (2) the optimality certificate and (3) the selection of the next active set if the validation of the optimality fails. In order to simplify our exposition and our comparison, we will start with a classical active set QP solver [7, p. 229] and a standard explicit solver (as proposed in [3], [2]). Corresponding author name Francesco Borrelli F. Borrelli is with Department of Mechanical Engineering, niversity of California, Berkeley, , SA fborrelli@me.berkeley.edu M. Baotić is with Faculty of Electrical Engineering and Computing, niversity of Zagreb, nska 3, 0000 Zagreb, Croatia mato.baotic@fer.hr J. Pekar is with Honeywell Prague Laboratory, Prague, Czech Republic jaroslav.pekar@honeywell.com G. Stewart is with Honeywell Automation and Control Solutions, North Vancouver, Canada greg.stewart@honeywell.com It is not the intent of this paper to compare the proposed algorithms with other very efficient explicit solvers appeared in the literature [8], [4], [], or fast QP solvers [], [6], [2], [5] tailored to the special structure of the underlying optimal control problem or suboptimal explicit solution. The comparison would be problem dependent and requires the simultaneous analysis of several issues such as speed of computation, storage demand and real time code verifiability. This is an involved study and as such is outside of the scope of this paper. II. NOTATION Throughout this paper (lower and upper case) italic letters denote scalars, vectors and matrices (e.g., A, a,...), while upper case calligraphic letters denote sets (e.g., A, B,...). R is the set of real numbers, N is the set of positive integer numbers. For a matrix (vector) A, A denotes its transpose, while A i denotes the i-th row (element). For a set A, A(k) denotes it s k-th element. Given the matrix G R m n, then for any set A {,..., m}, G A denotes the submatrix of G consisting of the rows indexed by A. Q 0 denotes positive definiteness (resp., Q 0 positive semidefiniteness) of a square matrix Q, while A denotes cardinality (number of elements) of a set A. III. CFTOC AND ITS STATE-FEEDBACK PWA SOLTION Consider the discrete-time linear time-invariant system subject to the constraints x(t + ) = Ax(t) + Bu(t) () E x x(t) + E u u(t) E (2) at all time instants t 0. In () (2), n x N, n u N and n E N are the number of states, inputs and constraints respectively, x(t) R nx is the state vector, u(t) R nu is the input vector, A R nx nx, B R nx nu, E x R n E n x, E u R n E n u, E R n E, and the vector inequality (2) is considered elementwise. Let x 0 = x(0) be the initial state and consider the constrained finite-time optimal control problem J (x 0 ) := J(x 0, ) x k+ = Ax k + Bu k, E x x k + E u u k E, k = 0,..., N where N N is the horizon length, := [u 0,..., u N ] R nun is the optimization vector, x i denotes the state at time i if the initial state is x 0 and the control sequence {u 0,..., u i } is applied to the system (), J : R nx R is (3)

2 the value function, and the cost function J : R nx R nun R is given as a quadratic function N J(x 0, ) = x N Q x N x N + x kq x x k + u kq u u k (4) k=0 where Q x = (Q x ) 0, Q u = (Q u ) 0, Q x N 0. Consider the problem of regulating to the origin the discrete-time linear time-invariant system () while fulfilling the constraints (2). The solution to CFTOC problem (3) (4) is an open-loop optimal control trajectory over a finite horizon. A Model Predictive Control (MPC) [0] strategy employs it to obtain a feedback control law in the following way: assume that a full measurement of the state x(t) is available at the current time t 0. Then, the CFTOC problem (3) (4) is solved at each time t for x 0 = x(t), and u(t) = u 0 is applied as an input to system (). A. Solution of CFTOC Consider the CFTOC problem (3) (4). By substituting x k = A k x 0 + k j=0 Aj Bu k j in (3) (4), this can be rewritten as the quadratic program [3] J (x) = 2 H + x F + 2 x Y x G b r + B x x where x = x 0, the column vector := [u 0,..., u N ] R n, n := n u N, is the optimization vector, H = H 0, and H, F, Y, G, B x, b r are easily obtained from Q x, Q u, Q x N and (3) (4) (see [3] for details). Because the problem depends on x the implementation of MPC can be performed either by solving the QP (5) on-line or, as shown in [3], [4], by solving problem (5) off-line for all x within a given range of values, i.e., by considering (5) as a multi-parametric Quadratic Program (mp-qp). In [3] the authors give a self-contained proof of the following properties of the mp-qp solution. Theorem : Consider the multi-parametric quadratic program (5) and let H 0. Then the set of feasible parameters X f is convex, the optimizer : X f R s is continuous and piecewise affine (PWA), and the value function J : X f R is continuous, convex and piecewise quadratic. Once the multi-parametric problem (5) is solved offline, i.e., the solution (x) = fpwa (x) of the CFTOC problem (5) is found, then the state-feedback PWA MPC law can be simply obtained by extracting first n u elements of fpwa (x) u (t) = [Inu 0nu... 0nu ] fpwa(x(t)). (6) Therefore by using a multi-parametric solver the computation of a MPC law becomes a simple piecewise affine function evaluation. IV. ACTIVE SET ALGORITHMS VS EXPLICIT SOLVERS The objective of this section is to compare computational time and storage demand associated to (i) active-set QPs for solving (5) and to (ii) the evaluation of the explicit solution (6). (5) Three main components are shared by active set QP algorithms and by explicit solvers: (i) the amount of off-line stored data, (ii) the validation of the optimality certificate and (iii) the selection of the next active set if the validation of the certificate fails. The three steps will be detailed later in this manuscript. For the sake of better readability we rewrite the quadratic program (5) compactly as 2 H + g(x) G b(x) where G R m n, b(x) R m, g(x) R n, b(x) = b r + B x x and g(x) = F x. Let I := {,..., m} be the set of constraint indices. For a fixed x, A ( x) denotes the set of active constraints at ( x): (7) A ( x) := {j I : G j ( x) = b j ( x)} (8) In the optimization field, the variety of QP algorithms is very rich and their performance depends on the type of problem. In this note, we prefer to highlight the main differences between the two approaches (online vs explicit) and corresponding changes in computational time and storage demand rather than selecting a specific QP implementation and carrying an exact computation. For this reason, in the next section IV-A, we present the main steps of a simple active set QP algorithm and in section IV-B we present the main steps of an explicit solver. For the same reason, we will not cover the variety of pivoting rules for degenerate case. A. Active Sets QP solver Before presenting an active set method for solving the QP (7), we will first consider a subset A of the constraints index A I and the following equality constrained QP: 2 H + g(x) G A = b A (x) A Lagrangian method [7, p. 229] solves the equality constrained QP (9) by computing a solution to the Karush Kuhn Tucker (KKT) conditions: H + G Aλ + g(x) = 0 G A = b A (x). (9) (0a) (0b) Equations (0a) and (0b) can be compactly written as [ ] [ ] [ ] H G A g(x) = () G A 0 λ b A (x) The matrix in () is referred as the Lagrangian matrix and it is symmetric. If the inverse exists and it is expressed as [ ] H G [ ] A L T = G A 0 T (2) S then the solution to () can be written as or equivalently as = Lg(x) + T b A (x) λ = T g(x) + Sb A (x) = (T B x A LF )x + T b r A = F Ax + c A λ = (SB x A T F )x + Sb r A = F d A x + cd A (3) (4)

3 The explicit expression for L, T and S when H exists are L = H H G A (G AH G A ) G A H T = H G A (G AH G A ) S = (G A H G A ) (5) In practice, computing equation (5) might not be numerically robust. Several alternatives can be found in [7, p. 236]. The matrix H is always invertible but the Lagrangian matrix in () might not be invertible. This happens when G A is not full row-rank. In this case any feasible point can be written as = Y b(x) + Zy, where y R n m, with m being the rank of G A. One possibility for computing the and λ is to use a QR factorization of the matrix G A to compute Y and Z as follows G A = Q [ R 0 ] = [Q Q 2 ] [ R 0 ] = Q R (6) where Q R n n is orthogonal, R R m m is upper triangular and Q R n m and Q 2 R n (n m). If we use Y = Q R and Z = Q 2 then [7, Eq. 0..4,0..5] =Y b(x) Z(Z HZ) Z (g(x) + HY b A (x)) λ =Y (H + g(x)) which combined with (4) gives: (7a) (7b) L = Z(Z HZ) Z, T = Y LHY, S = Y HT (8) Alternative approaches for computing and λ can be found in [7]. We remark that they all consists in manipulating () and, after a certain number of operations, obtaining (x) and λ (x). We are now ready to introduce a primal feasible QP solver. Consider the QP (7) for a fixed x, the set A = A(x) of active constraints at x and the associated KKT conditions: H + G Aλ + g(x) = 0 G A = b A (x), λ 0, G I\A < b I\A (x). (9a) (9b) (9c) (9d) Next the main ingredients a primal active set method are briefly recalled [7]. At each iteration k, a feasible point (k) for the QP (7) is known with associated active constraint set A (k) := {j I : G j (k) = b j (x)}. Step k consists of computing the solution to the problem where δ 2 δ Hδ + g (k) (x)δ G A (k)δ = 0 (20) g (k) (x) = g(x) + H (k) (2) and δ represents a correction to (k) in the direction where the constraints A (k) are active. Problem (20) is an equality constrained QP which can be solved as shown above. Let δ be the optimizer of problem (20). Three cases are possible: ) δ 0 and (k) + δ is feasible for (7). Then, set (k+) = (k) + δ and iterate the procedure with (k+) and A (k+) = A (k). 2) δ = 0. (k) might be the optimal solution or might violate duality conditions. The Lagrange multipliers λ (k) associated to (k) and its active set A (k) by using (7b)-(8) are computed together with the smallest multiplier d = i {,..., A (k) } λ (k) i and its index p = arg i {,..., A (k) } λ(k) i. If d > 0, then all Lagrange multipliers are positive and the optimum is found, otherwise set A (k+) = A (k) \ A (k) (p) and iterate the procedure. 3) (k) + δ is not feasible for (7). Then find the best feasible point (k+) = (k) + δ (k) with a line search in the direction of δ, i.e., δ (k) = α (k) δ where α (k) is computed so that no constraint is violated: α (k) = b i G i (k) i I\A (k), G iδ >0 G i δ (22) At (k+) a new constraint becomes active and it is defined by the index (p say) which achieves the in (22). Set (k+) = (k) + α (k) δ and A (k+) = A (k) p and iterate the procedure. The algorithm can be summarized as follows Algorithm :. Given () and A (), set k = 2. if δ = 0 is not a solution of (20)-(2) THEN GOTO Step Compute λ (k) by using equations (7b),(8) and solve d = λ (k) i {,..., A (k) i (23) } 4. IF d > 0 THEN set = (k) and terate 5. ELSE A (k) = A (k) \ A (k) (i d ) where i d is the arg of problem (23) 6. DO QR decomposition of G A (k) as in (6) and set Z (k) = Q 2 7. Solve the following 2 δ Z (k) HZ (k) δ + g (k) (x)δ (24) with g (k) (x) = (g(x) + H (k) ), which yields 8. Compute α (k) = δ = (Z (k) HZ (k) ) g (k) (x) (25) (, i I\A (k), G iδ >0 b i (x) G i (k) ) G i δ (26) and set (k+) = (k) + α (k) δ 9. IF α (k) < then A (k+) = A (k) i p where i p is the arg of problem (26) 0. Set k=k+ and GOTO 2. We can characterize Algorithm based on the three following elements: Off-line Storage Matrices H, F, G, B r and b x. Optimality Certificate (k) is optimal if primal feasibility and dual feasibility are satisfied.

4 Active Set Selection Algorithm proceeds by checking dual feasibility first. If it is not satisfied, then the constraint which is violated the most in (23) is included in the active set and procedure is repeated. If dual feasibility is satisfied, then primal feasibility is verified. If primal feasibility is not satisfied then (26) is solved to exclude the constraint which violates primal feasibility the most. Alternative Simple Algorithm The algorithm presented in this section is not typically used to solve QPs. It is introduced to better understand the issues involved with the explicit algorithms presented later in this paper. Compared to Algorithm, the algorithm proposed in this section does not require a feasible initial point and does not provide primal feasible solutions at intermediate steps. At each iteration k, an active constraint set A (k) is known and the equations (0) are solved to obtain (k) (x) and λ (k) (x). The algorithm proceeds as follows ) If (k) is not feasible for the QP (7), then compute the constraint that is violated the most i p = arg i I\A (k)b i (x) G i (k) (27) Set A (k+) = A (k) i p and iterate the procedure. 2) Compute the smallest multiplier d = i {,..., A (k) } λ (k) i and its index i d = arg i {,..., A (k) } λ(k) i. If d > 0, then all Lagrange multipliers are positive and the optimum is found, otherwise set A (k+) = A (k) \ A (k) (i d ) and iterate the procedure. The two steps above are the main components of the algorithm summarized below Algorithm 2:. Given A (), set k = 2. Compute (k) by using equation (7a), (8) 3. Compute f = b i (x) G i (k) (28) i I\A (k) 4. IF f < 0 then A (k+) = A (k) i p where i p is the arg of problem (28), i.e. (27). Set k=k+ and GOTO Compute λ (k) by using equation (7b),(8) and solve d = λ (k) i {,..., A (k) i (29) } 6. IF d > 0 THEN set = (k) and terate 7. ELSE A (k) = A (k) \ A (k) (i d ) where i d is the arg of problem (29) 8. Set k=k+ and GOTO 2. Remark 4.: We remark that Off-line Storage, Optimality Certificate and Active Set Selection for Algorithm are identical to Algorithm 2. The only difference between Algorithm and Algorithm 2 is that the first guarantees primal feasibility of the intermediate solution at each step while the latter does not. This is obtained at the price of an initial feasible solution required to initialize Algorithm and the additional steps in (26) required to compute a feasible solution. B. Explicit Solution: Off-line and On-line Computation The basic idea beyond the computation of the explicit solution to QP (7) for a set of parameters x K R nx can be described as follows. Consider an active set A k and the primal and dual solution, λ in (3)-(5) if the Lagrangian matrix is invertible or (3), (6)-(8) if Lagrangian matrix is not invertible. and λ are affine functions of x since b(x) and g(x) are affine functions of x. Their expression is valid for all x for which the set A k is active at the optimum (note that in this section the lower index k in A k simply denotes a counter, while in the previous section, the upper index (k) specifies variables at the k- the step of the algorithm). Such region CR Ak is called the critical region and is the set of all parameters x for which the constraints indexed by A k are active at the optimizer of problem (7). The critical region CR Ak is computed by imposing primal feasibility conditions (9d) on P p := {x : G I\Ak (x) < b I\Ak (x)} (30) and dual feasibility conditions (9c) on λ P d := {x : λ (x) 0} (3) In conclusion, the critical region CR Ak is the intersection of P p and P d : CR Ak = {x : x P p, x P d } (32) Obviously, the closure of CR Ak is a polyhedron in the x- space. Based on (4) we can rewrite the closure of the primal and dual polyhedron (30)-(3) as: P p = {x : G I\Ak F Ak x b(x) I\Ak G I\Ak c Ak } = {x : P i,p x Q i,p, for i =,... I \ A k } (33) and P d := {x : F d A k x c d A k } = {x : P i,d x Q i,d, for i =,... A k } (34) An mp-qp algorithm deteres the partition of K into critical regions CR Ai, and finds the expression of the functions ( ) for each critical region. Let N P denote the total number of critical regions. The explicit solution (4) of the QP (7) is (x) = F Ai x + c Ai, x CR Ai, i =,..., N P, (35) where F Ai R n nx, c Ai R n, {CR Ai } N P i= is a polyhedral partition of K (i.e., i CR Ai = K, and CR Ai and { CR Aj have disjoint interiors i j), with CR Ai = x R n x P i x Q i}, P i R pi nx, Q i R pi, and p i is the number of halfspaces defining polyhedron CR Ai, i =,..., N P. Note that in general i=,...,n P CR Ai = K K since for some x the QP (7) could be infeasible. In principle, one could simply generate all the possible combinations of active sets and compute the corresponding

5 critical region CR Ai and optimizer expression of ( ). However, in many problems only a few active constraints sets generate full-dimensional critical regions inside the region of interest K. Therefore, the goal of an mp-qp solver is to generate only the active sets A i with associated fulldimensional critical regions covering only the feasible set K K. The evaluation of explicit solutions in its simplest form would require: (i) the storage of the list of polyhedral regions and of the corresponding affine control laws, (ii) a sequential search through the list of polyhedra for the i-th polyhedron that contains the current state in order to implement the i-th control law. Since verifying if a point x belongs to a critical region means to verify primal and dual conditions, then the on-line search for the polyhedron containing x can be compared to the main steps of a QP solver. Next Mj i denotes the j-th row of the matrix M i. The simplest implementation consists of searching for the polyhedral region that contains the state x(t) as in the following algorithm: Algorithm 3:. i = 0, notfound=tre 2. WHILE i N P AND notfound 2.. j = 0, feasible=tre 2.2. WHILE j p i AND feasible IF Pj ix(t) > Qi j THEN feasible=false 2.3. END 2.4. IF feasible THEN notfound=false 3. END We can characterize Algorithm 3 based on the three following elements: (i) Off-line Storage: Matrices F Ai, c Ai, P i, Q i, (ii) Optimality Certificate: Primal feasibility and dual feasibility, (iii) Active Set Selection: Pick the next active set in the list. Algorithm 3 differs from the primal feasible Algorithm 2 in the way the next active set (or region) is chosen. One can easily modify Algorithm 3 to use the same active strategy of Algorithm 2 as follows. In the j-th critical region CR Aj we separate the constraints deriving from primal feasibility P j,p x Q j,p and the constraint deriving from dual feasibility P j,d x Q j,d. In the next algorithms A (k) (p) denotes the p element of the set A (k) and the symbol j A (i) p denotes the following operation: given the active set A (i) at step i, select the index j so that A j = A (i) p and set A (i+) = A j. Algorithm 4:. notfound=tre, k =, j =, A () = A. 2. WHILE k N P AND notfound 2.. Compute f = i P j,p i x(t) + Q j,p i and let i p be the arg 2.2. IF f < 0 THEN j A (k) i p, k = k + break 2.3. Compute d = i P j,d i x(t) + P j,d i and let i d be the arg 2.4. IF d < 0 THEN j A (k) \ A (k) (i d ), k = k + break 2.5. notfound=false 3. END Algorithm 4 can be interpreted as the off-line version of Algorithm 2. The off-line version of Algorithm can be found in [5]. V. COMPARISON BETWEEN ON-LINE AND EXPLICIT ALGORITHMS By comparing Algorithm 2 and Algorithm 4 we can draw the following conclusions. (I) The active set strategy and the certificate of optimality are the same. In particular, Step 5. in Algorithm 2 corresponds to Step 2.. in Algorithm 4. The difference is clear from equation (34). In the explicit Algorithm 4 the matrices Pj,d x and P j,d are precomputed and stored. In the online QP Algorithm 2, Pj,d x and P j,d are computed online from (3) and (8). Similarly, Step 3. in Algorithm 2 requires the computation of b i (x) G i (k) which corresponds to Step 2.3. in Algorithm 4. The difference is clear from equation (33). In the explicit Algorithm 4 the matrices Pj,p x and P j,p are precomputed and stored. In the online QP Algorithm 2, Pj,p x and P j,p are computed online from (3) and (8). Although equations (4) and (8) can be efficiently computed, online QR decomposition and several matrix multiplications are required at each time step. This represents the main computational saving between online active sets QPs and explicit solvers. (II) The online QP Algorithm 2 could be improved. Any modification which helps computing the L, T, and S in (8) will reduce the computational gap between Algorithm 2 and Algorithm 4. However, a computational gap will always exist unless those matrices are pre-computed as in Algorithm 4. Any modification on the active set strategy or to the optimality certificate can be applied to the explicit solution as well and therefore they will not affect the comparison. (III) Algorithm 2 and Algorithm 4 can be properly compared only if they are initialized with the same set A (). Note that there might be active constraint sets for which it is possible to initialize Algorithm 2 and not Algorithm 4. In fact, Algorithm 4 stores only full dimensional critical regions for which the corresponding set A (k) is active at (x) for some x, i.e., A (x) defined in (8). Definition (Non-degenerate QP): We say that the QP (7) is non-degenerate if for each x K the rows of G A (x) are linearly independent, where A (x) is defined in (8). The following proposition shows that if Algorithm 2 and Algorithm 4 are initialized with the same set A () then they explore the same active sets. Proposition : Assume the QP (7) is non-degenerate, let x be a feasible state and consider Algorithm 4. If A () corresponds to a full-dimensional critical region stored offline by Algorithm 4, then, at any time step k, either j A p or j A \ A(p) correspond to full-dimensional critical regions which have been stored off-line by Algorithm 4. At the generic step k of Algorithm 4, assume that constraint j of the current critical region CR A (k) (say a j x b j ) is violated. Let A (k+) A (k) p ( A (k+) A (k) \

6 A (k) (p)) the new set of active constraints if j is a primal (dual) constraint. Then, three cases can occur: () CR A (k+) has been stored off-line by Algorithm 4, (2) CR A (k+) is not a full dimensional critical region and (3) CR A (k+) is empty. Case (2) can be excluded because we assumed that the QP (7) is non-degenerate (cf. [4], [3]). Case (3) can also be excluded by contradiction. In fact, assume there is no critical region in the half-space a j x > b j. Then a j x b j is a facet of the region of feasible states K and therefore (since it is violated) x is not feasible, which contradicts the assumption. Therefore, the only admissible option is case () which proves the proposition. (IV) When computing the critical regions (32) redundant constraints are removed. While, in general, this can improve the efficiency of Algorithm 4 versus Algorithm 2, by definition these constraints will not play any role in selecting the next set of active constraints. (V) From the above observation it is clear that Algorithm 2 requires more operations at each iteration than Algorithm 4. This is obtained at the price of increased memory requirement. In fact, in Algorithm 4 the polyhedral partition and the gains have to be stored which, in general, largely surpass the memory required for Algorithm 2 (simply the matrices of the QP (7)). Remark 5.: If the QP (7) is degenerate, then QP Algorithm 2 has to be modified in order to avoid (possible) cycling. There are standard well know pivoting approaches which solve this issue. Consequently, the selection of the next region in the explicit Algorithm 4 has to be modified accordingly. Remark 5.2: The operation j A (k+) is immediate in Algorithm since it is a simple selection of different matrix rows. In Algorithm 4, if the list of neighboring regions is available, then j A (k+) is not time consug (since it corresponds to switch to the neighboring region of a given facet). However, the construction of neighboring region list can be a numerically sensitive issue for mpqp solvers especially in the case of degeneracies. If the list of neighboring regions is not available then, the operation j A (k+) requires a search through a list of active constraints associated to all the stored region and it might be time consug. Remark 5.3: The comparison of Algorithm 3 and Algorithm 2 is similar to the comparison between Algorithm 2 and Algorithm 4 with two main differences. Algorithm 3 corresponds to an active set QP where the next active set is chosen randomly. Although, on average it might perform worse than Algorithm 4, it does not require the computation of all dual variables as in step (2.3.) of Algorithm 3 and of all primal feasibility conditions in as step (2..) of Algorithm 4. As soon as one is violated, the algorithm moves to the next region in the list. This does reduce the computational time of each step. Secondly, Algorithm (3) works well even in presence of non full-dimensional critical regions (in fact it does not require the list of neighboring regions). For these reasons, Algorithm (3) is very simple and practical even if it might perform very poorly on average. VI. CONCLSIONS We have shed some light on the complexity of the on-line solution of active-sets quadratic programs versus the on-line evaluation of explicit solutions. Three elements can be used to compare the different algorithms: () the amount of stored data, (2) the validation of the optimality certificate and (3) the selection of the next active set if the validation of the certificate fails. If the algorithms are properly initialized, the main difference between the two approaches relies on the choice between the online QR decomposition of a set of linear equations and their off-line solution. In the latter case computational time is gained at the price of memory storage. 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