The continuous quadrant penalty formulation of logical constraints

Could continuous optimization address efficiently logical constraints? We propose a continuous-optimization alternative to the usual discrete-optimization (big-M and complementary) formulations of logical constraints that leads to an effective practical method. Based on the simple idea of guiding the search of a continuous-optimization method towards the parts of the domain where the logical constraint is satisfied, we introduce a smooth penalty-function formulation of logical constraints, and related theoretical results. The effectiveness of this formulation of logical constraints, that allows the direct use of state-of-the-art continuous optimization solvers, is demonstrated on an aircraft conflict avoidance application.


Introduction
Logical constraints, such as an alternative constraint (t(x) ≤ 0 or f (x) ≥ 0) and a conditional constraint (t(x) > 0 implies f (x) ≥ 0), are common in mathematical optimization models.Mathematical optimization practitioners typically reformulate a logical constraint at the expense of the introduction of one binary auxiliary variable, to get either a big-M formulation or a complementary formulation.The former involves constraints that are linear but that depend on a large-enough constant yielding potential numerical instability/inefficiency, while a complementary formulation involves nonlinear constraints.We consider whether continuous optimization could play a role in this context.We propose a novel modeling approach, the continuous quadrant penalty formulation, based on a continuous optimization penalty function to handle logical constraints without requiring the introduction of binary variables.Our use of penalization is also original in that it does not directly result from adding some measure of the violation of a constraint (difference between the right-hand side and the left-hand side values) in the objective function.More precisely, the penalty function we are introducing is not the usual monotonically non-decreasing function of the distance between a given point and the desirable set.We shall see that this distance is not differentiable; neither is the square of this distance.However, our penalty function can be seen as a smooth approximation of the square of the distance from the desirable set.
The remainder of this paper is structured as follows.Section 2 is a short introduction to logical constraints and penalty-function approaches.Section 3 proposes a simple piecewise-linear penalty function to model logical constraints together with some theoretical results.In order to get round the difficulties related to addressing non-differentiabilities, Section 4 then demonstrates the existence of, and constructs explicitly, a smooth penalty function which is piecewise quadratic.Section 5 reports computational tests on an aircraft conflict avoidance application to show the effectiveness of our approach.Finally, conclusions are drawn in Section 6.

Logical constraints and penalty functions
Conditional constraints have the form for some real-valued functions t and f of x ∈ R n .Such logical constraints, together with feasibility constraints, alternative constraints and compound alternatives are common in operations research.They are typically modelled using mixed-integer formulations (see for instance [3]).The conditional constraint (1) is logically equivalent to the alternative constraint: (at least one condition must be satisfied) since given two logical propositions P and Q, one has that (P ⇒ Q) is logically equivalent to (¬P ∨ Q), i.e., non P or Q).
The usual approach to model (2) relies on the so-called big-M formulation: (2) is modelled by alternative constraints, involving the introduction of a binary 2 decision variable y, as follows: where the big-M constants M 1 and M 2 are chosen large enough so that one has: The decision y = 0 corresponds to imposing the constraint t(x) ≤ 0, while y = 1 enforces the constraint f (x) ≥ 0 to be satisfied.Best results in terms of computational times and numerical stability are expected when the constants M 1 and M 2 are chosen as small as possible to satisfy (4).
The other mathematical modeling trick to deal with the alternative constraint (2), called complementary formulation, simply yields the constraints: where, again, y is an extra auxiliary binary decision variable.The authors of [1] claim that this complementary formulation can be addressed in a more efficient way than the big-M formulation for certain applications involving functions t and f that are linear, as the latter formulation yields weak continuous relaxations while the former allows one to implement tighter bounds to be exploited by MINLP solvers.Remark that obtaining good continuous relaxations (studied, e.g., in [2] in the general context of disjunctive programming) is out of the scope of the present paper, which rather introduces a continuous-optimization formulation.
In the following, we propose an original continuous-optimization model to deal with the conditional constraint (1) that is based on penalization.
The original motivation of penalty functions was to replace difficult constrained optimization problems by a sequence of easier unconstrained optimization problems for which we had good algorithms.There are close relations between penalty methods and numerous nonlinear optimization techniques (e.g., augmented Lagrangian, successive quadratic programming, interior point methods).The pioneering work of Conn in penalty methods dates back to 1971 in his doctoral thesis [7] and the introduction of a method described in [8] to deal with the non-differentiable penalty function: (where µ ≥ 0 is a weighting penalty parameter ) resulting from the method of replacement [15] of Pietrzykowski (Conn's PhD advisor) to address constraints of the form: Conn concludes his 1981 NATO-conference paper on penalty function methods [9] by asking "whether penalty functions have any significant contributions to make to discrete or global optimization".In [14] Gamble, Conn and Pulleyblank specialize a non-linear penalty function method to a combinatorial algorithm for the minimum cost network flow problem.
The main contribution of the current paper is the proposition of a continuousoptimization alternative to the usual discrete-optimization -big-M (3) and complementary (5) -formulations of logical constraints, that relies on a penalty function.The intuition behind the continuous quadrant penalty formulation of the logical constraint we introduce is an attempt to guide the search of a constrained continuous-optimization method towards the parts of the domain where the logical constraint (1) is satisfied.Remark that in this context only the logical constraints will be penalized, as other types of constraints (such as those of the application we consider in Section 5) can readily be addressed by state-of-the-art continuous optimization methods.
Let us consider, for any given decision-variable vector value x ∈ R n , a couple, (t(x), f (x)) ∈ R 2 .In the sequel, we shall drop the explicit dependency upon x for simplicity of the presentation.Imposing the conditional constraint (1) is therefore equivalent to requiring that the point p := (t, f ) lies in the non-convex set R 2 \ S, where S is the open fourth quadrant: as illustrated in the grey areas in Figure 1.Thus, one can obtain a continuous optimization formulation avoiding the introduction of the binary variable y by replacing the conditional constraint with the constraint: It is this constraint that we propose to model by the use of a penalty approach.
To this aim, let us design an appropriate penalty function, g : R 2 → R. It is desirable to consider candidate functions g that are continuous, possibly even smooth, and such that a descent method minimizing g will converge towards a point in R 2 \ S.
Let us first introduce a definition and a proposition that will be useful in the sequel.
The indicator function 1 S (whose value is one in S, and zero elsewhere) is a special case of S-violation dipsticks.Other examples are each of the terms of the classical ℓ 1 penalty function: corresponding to the constraints (7).Indeed, given i ∈ I, the term max(0, g i (x)) is an S-violation dipstick, considering S to be here the subdomain where the constraint g i (x) ≤ 0 is violated (and similarly for i ∈ E and the term |g i (x)|).
Proposition 1.Let S ⊂ R n be such that R n \ S is not convex.Then, no S-violation dipstick function can be convex.
Proof.Let g : R n → R be an S-violation dipstick function.Since the set R n \ S is not convex, there exist necessarily two points x, y ∈ R n \S, x ̸ = y, and a scalar 0 < λ < 1 such that the point λx + (1 − λ)y lies outside R n \ S, i.e., in S. Thus, g(λx+(1−λ)y) > 0 by Definition 1 b).Moreover, by a), λg(x)+(1−λ)g(y) = 0, since both x and y lie in R n \ S. Thus, g is not a convex function.
In particular, the above proposition says that an S-violation dipstick cannot be a linear function when R n \ S is not convex.One must therefore search for a somewhat more complex penalty function.

A piecewise-linear penalty function
A first natural choice beyond linearity is that of a piecewise-linear function.The next proposition states that at least three pieces are then needed for designing such a penalty function.
Unless S is a half-space, there does not exist a continuous two-piece piecewise-linear S-violation dipstick.
Proof.Let g : R n → R be a continuous two-piece piecewise-linear S-violation dipstick.The function g must then be linear over S with value zero at the frontier of S. This is not possible unless S is a half-space or g is identically zero.
Let us come back to the case of interest for us, where n = 2 and S is the open fourth quadrant.From the two above propositions, we know that if we want to design a continuous S-violation dipstick penalty function that is piecewise linear, it must have at least three pieces (since R 2 \ S is not convex and, a fortiori, not a half-space).Proposition 3. Let S ⊆ R 2 be the open fourth quadrant.There exists a continuous piecewise-linear S-violation dipstick, noted g α , having exactly three pieces: where α > 0 is any given positive (slope) parameter.Moreover, up to a multiplicative constant, and up to the arbitrary value of α, this function is unique.
The values of g α over its three pieces are displayed on Figure 1.Remark that g α can also be expressed as the one-line formula: Proof.Regarding the existence part, one can easily verify that g α is a continuous three-piece piecewise-linear S-violation dipstick function.
As for the unicity of g α , first observe that S is necessarily partitioned into two pieces, over each of which g α is linear, and the common frontier of these two pieces must be a straight-line segment (otherwise, g α could not possibly be linear over each of theses two pieces).Moreover, this straight-line segment frontier must go through the origin.Indeed, if this were not the case, as in the proof of Proposition 2, g α would be zero over one of the two S pieces which, in turn, would yield an identically-null function g α .Without loss of generality, let f = −αt be the equation of the corresponding straight line, where α > 0. Further, let g α1 and g α2 be respectively the restriction of g α over each of the two S pieces.One can show (see [11, Proposition 1]) that for g α to be continuous, the difference of gradients,∇g α1 − ∇g α2 , is necessarily a (non-zero) multiple of (α, 1) T (the gradient of the ridge: {(t, f ) : f = −αt)}).This, together with the fact that g α1 and g α2 are linear functions satisfying g α1 (0, f ) = 0 for all f < 0 and g α2 (t, 0) = 0 for all t > 0 yields the continuous piecewise linear function whose gradients are displayed on Figure 1 (right), up to some positive constant.The only such piecewise linear function is, up to this multiplicative constant, the proposed function g α .
Definition 2 (function leaning outwards S).Let ∅ ̸ = S ⊊ R n .A function g : R n → R is said to lean outwards S if for any given point p ∈ S, and for any descent direction d for g at p, there exists a threshold step size γ > 0 such that p + γ d ̸ ∈ S, for all γ ≥ γ.
Let us consider again the example of the classical ℓ 1 penalty function (11), this time in the simple special case of the following 2D quadratic constraint: i.e., g(x) = max(0, c(x)).This function g does not lean outwards the subdomain, S = {x : c(x) > 0}, where the constraint is violated.To see this, consider ).The point p is in S, the steepest-descent direction for g at p is d = −(3, 2 √ 7), and for whatever step size, γ > 0, chosen one can show that p + γ d ∈ S.An even simpler manner to see this here is to observe that there cannot be any penalty function that can lean outwards a set S when the complement of this set is bounded.
It seems therefore rather demanding to require a penalty function to lean outwards the infeasible domain.However, the piecewise-linear penalty function g α introduced in Proposition 3 does lean outwards the forbidden area S defined by (8), as this will be demonstrated below (Proposition 4).
Definition 3 (S-repulsive penalty function).Let ∅ ̸ = S ⊊ R n .A function g : R n → R is an S-repulsive penalty function if it satisfies the following three requirements: iii) g is continuous Proposition 4. Let S ⊆ R 2 be defined by (8).The piecewise-linear function g α defined by ( 12) is an S-repulsive penalty function.
Finally, for the case where g α is not differentiable at p = ( t, f ) (i.e., f = −α t), the set of all possible descent directions is: Thus, one necessarily has either d1 < 0 or d2 > 0. It suffices to set γ = − t d1 in the first case, and γ = − f d2 in the second case, to obtain: p + γ d ̸ ∈ S, for all Figure 2: g α (left) and g 2 α (center) for α = 1, restricted to the line (in blue on Figure 3, left) f = t − 2X, for some X > 0. Right: g 2 α (black) and g β (blue) when β approaches 1.
Let us consider for the remaining of the paper the case where n = 2 and S is the open fourth quadrant, relevant for addressing the logical constraint (9).
Although the piecewise-linear function g α has several desirable properties as a candidate for a penalty function (it is S-repulsive), it is not smooth along the three straight-line-segment frontiers: t = 0, f = 0 and f = −αt.For the sake of simplification, let us focus on the case where α = 1.
The non-differentiabilities of g α correspond to the three points of discontinuity of the restriction of g α (displayed on Figure 2, left) to the line f = t−2X (for some given X > 0): at the abscissas ± √ 2X and at 0. This non-smoothness is a drawback when selecting a state-of-the-art optimization method for addressing a penalty-function-based model.Special-purpose optimization methods could be designed for such structured non-differentiable objective functions, as it is done for instance in [10,11], but we are here focusing on addressing logical constraints with standard continuous optimization approaches.Note that, contrary to the common approach of squaring the terms of the ℓ 1 -penalty function (11) in order to smooth out the non-differentiabilities, here one can easily verify that along the half-line f = −αt with t > 0, the function g 2 α is not differentiable (see Figure 2, center).A smoothing technique such as that proposed by Pinar and Zenios [16] that replaces a max(0, f ) term with a (three-piece) piecewisequadratic function of f is not appropriate either according to Proposition 1, as their technique can be applied only to convex functions.

A smooth piecewise-quadratic penalty function
Let us construct a smooth penalty function, so that efficient state-of-the-art methods can be applied to the resulting optimization problem.We choose here to build a piecewise-quadratic approximation (that we shall name g β ) of g 2 α (our non-convex piecewise-linear penalty function, g α , squared).
One can easily infer that a smooth S-violation dipstick penalty function that is piecewise quadratic will have at least four pieces.Indeed, simply consider the restriction of such a function, say g, to the line f = t − 2X, for some positive constant X, as displayed on Figure 3 (center): it is easy to show that there is no way to join smoothly any two quadratic polynomials p 1 and p 3 whose value and derivative are zero at the intersections of the line and the axes; a third quadratic polynomial, p 2 , is needed.Three pieces are therefore required over the fourth quadrant, plus one piece where g is identically zero in the remaining quadrants.
Proposition 5 (Existence).Let S ⊆ R 2 be the open fourth quadrant.There exists a smooth S-repulsive penalty function, g β : R 2 → R, that is piecewise quadratic with exactly four pieces.A family of such functions is, for β ∈ R, β > 1: Before proving this result, let us visualize this 2D function, g β , for a particular value of the parameter: β = 3.It takes the value zero outside the fourth quadrant, and is composed of three quadratic pieces in the fourth quadrant.Figure 4 displays the values g β takes over its four subdomains (left), and the 3D graph of g β (right).Figure 3 displays the restriction of g β to a transverse line f = t − 2X, for some given value of X > 0 (center), and the restriction of g β along its axis of symmetry: the line f = −t (right).Observe that along any ray into S, the restriction of g β is a convex parabola whose minimum corresponds to (0, 0), as shown on Figure 3 (right).
Proof.Let β > 1.We shall first show that g β is an S-violation dipstick, then that g β is a smooth function, and finally that g β leans outwards S.
In order to show that g β is an S-violation dipstick, consider a point p = (t, f ) ∈ R 2 .If p ̸ ∈ S, then by definition of g β , g β (p) = 0.If p pertains to either subdomain where g β (t, f ) = t 2 or f 2 , then one clearly has g β (p) > 0. In order to prove that g β is positive on the remaining subdomain, where consider the univariate function h defined as the restriction of g β to the line, illustrated on Figure 3 (right): f = t − 2X, for some given value X > 0 (any given point ( t, f ) pertains to such a line: simply take X = t− f 2 ).One has: Observe that on that line, (14) β+1 , the univariate concave quadratic function h is minimized either at t = 2X (β+1) or at t = 2Xβ (β+1) .Evaluating h at these points, one obtains: (β+1) 2 > 0 and similarly, one finds h( 2Xβ β+1 ) = 4X 2 (β+1) 2 > 0. Therefore, g β is also positive on the domain where ( 14) is satisfied, and g β is therefore an S-violation dipstick.

Remarks:
1.The piecewise quadratic function, g β , is not unique.It is defined up to a multiplicative constant, and changing the units of t and f (scaling) amounts to changing the slope of its axis of symmetry (here set to f = −t).
2. As for classical penalty functions, g α (in the case where α = 1) is a monotonically non-decreasing function of the (ℓ 2 ) distance between a given point and the desirable set, in our case the non-convex set R 2 \ S. Let us denote this distance by d R 2 \S .In contrast, it is easy to show that the penalty function g β (with β > 1) is not a monotonically non-decreasing function of d R 2 \S .This is illustrated on Figure 5 where, for a smallenough positive step size γ > 0: 3. When α = 1 and β takes a value near 1, the angle of the sector − f β < t < −βf (over which g β takes the values of the concave quadratic function 1  1−β 2 (t 2 + 2β t f + f 2 ) -see Figure 4, left) becomes small, and g β approaches the three-piece piecewise quadratic function g 2 α (obtained by squaring the piecewise linear function g α ).The function g β can therefore be seen as a smooth approximation of g 2 α (for the symmetric case where α = 1).In fact, both functions coincide, except on the central sector, as illustrated on Figure 2 that plots the restrictions of g 2 α (for α = 1, center) and of g β (right) along the transversal line f = t − 2X (X > 0) for a value 4. In order to guide a user to choose a particular value of β, let us consider again the restriction, h X , of g β to the line f = t − 2X, for some given X > 0. This univariate piecewise-defined function h X is composed of five pieces: one straight-line (identically zero), three univariate quadratic polynomials p X 1 , p X 2 , p X 3 , and again one straight-line (identically zero).The case where each of the four breakpoints (discontinuities in the second derivative of h X ) together with the point corresponding to the axis of symmetry of h X are equally spread over the domain where h X is positive, corresponds to the special case where β = 3, and is illustrated on Figure 3.

Computational assessment
This section aims at demonstrating the usefulness of the quadrant penalty formulation of logical constraints for real-world optimization problems.We address in particular the aircraft conflict avoidance problem, which involves several logical constraints, as many as there are possible pairs of aircraft at stake.The aim is to keep aircraft pairwise separated by a given separation distance, d, (otherwise, there is a conflict).The inherent combinatorics in such conflict avoidance problems leads naturally to conditional constraints: if two vehicles are converging spatially, then their smallest inter-distance must be bounded below at all time.The usual big-M or complementary formulation approaches to address these logical constraints require introducing as many binary variables as there are pairs of aircraft.We show that our quadrant penalty formulation of logical constraints allows continuous optimization to make a significant contribution to solving such problems that are typically addressed by combinatorial optimization approaches (see [4] and references therein).To that aim, we consider here more specifically the continuous optimization formulation of this problem, presented in [5].
In the variant of the aircraft conflict avoidance problem that is the closest to air transportation operational constraints, both the speed and the heading angle of each aircraft can be changed to separate the trajectories (see e.g., [12]).Let A = {1, 2, . . ., n} be an index set corresponding to aircraft flying at a same given altitude, on straight-line segment trajectories.The following input data are given for each aircraft i ∈ A at a certain time of interest t = 0: initial position, (x 0 i , y 0 i ) ∈ R 2 ; initial speed, v i ; and initial heading angle, ϕ i .The aim is to decide speed and heading-angle variations for each aircraft i, simultaneously, to guarantee that aircraft remain pairwise separated.In other words, considering a particular pair of aircraft i, j ∈ A (i < j), the following separation constraint must be satisfied: where the vector x ij (t) = x i (t) − x j (t) ∈ R 2 gives the position of aircraft i relative to that of aircraft j, and where || • || stands for the usual, Euclidean norm.This constraint not only induces in general a non-convex feasible set, but it must moreover hold at all time t ≥ 0. Following [6], one can reformulate this semi-infinite constraint (defined over a continuous time interval) by first re-writing it as the univariate quadratic constraint: where v ij is the velocity vector of aircraft i relative to that of aircraft j, and x 0 ij is the vector of their relative initial positions.Second, supposing that aircraft are separated at t = 0 (otherwise the problem can straightforwardly be detected as an infeasible problem), one can suppose the non-trivial case where v ij ̸ = 0. Third, one remarks that ( 17) is satisfied if and only if it is satisfied at the unique time t = t m ij that minimizes the (strictly convex) function f ij .This time t of maximal proximity between aircraft i and j is easily computed as Finally, replacing t = t m ij in the separation constraint (17) yields the logical constraint equivalent to (16): where The aircraft conflict avoidance problem boils down to a feasibility problem that aims at deciding speed variation, q i , and heading angle variation, θ i maneuvers for each aircraft i ∈ A so that the separation constraints are satisfied.More precisely, once these decisions are made, the new speed of aircraft i is q i v i , and its new heading angle is ϕ i + θ i , where v i is the aircraft original speed, and ϕ i is its original angle.Following [5], for the angle decision, we rather consider the heading-angle maneuver variables, ω i , π i , that are related to both the original angle decision variable, θ i , and to the speed decision variable, q i , as follows: The constraints to be satisfied are: where v ij , t m ij , and f m ij are auxiliary decision variables that can be straightforwardly computed explicitly in terms of the decision variables: q i , ω i , π i and q j , ω j , π j and of the input data, and where q i , q i , ω i , ω i , π i , π i are given lower and upper bounds on the main decision variables.
Modeling the logical constraints (21) through the continuous quadrant penalty function, g β , introduced in Proposition 5, yields an optimization problem whose objective function, to be minimized, reads: subject to constraints ( 22)-(28).This allows the direct use of state-of-the-art continuous optimization solvers.
Remark that the objective function of this optimization problem is not a convex function (by Proposition 1), and the feasible domain ( 22)-( 28) is not a convex set, which can be seen simply by considering constraints (25).A mathematical optimization method for continuous optimization may thereby converge to a local minimum that does not satisfy the original (penalized) logical constraints (non-zero optimal value), yielding a so-called local infeasibility.
We present numerical results obtained considering benchmark instances known as Random Circle Problems (RCP) and available from [12].These instances are generated by positioning aircraft uniformly on the circumference of a circle, which represents the observed airspace portion and whose radius is 200 nautical miles (NM), and assigning to each aircraft a randomly-generated initial speed (in the range 486-594 NM/h) and a heading (deviated with respect to the heading towards the center by an angle in the range [−π/6, π/6]) to fly towards the opposite side of the circle.This results is realistic instances.We consider three different values for the number, n, of aircraft, namely n = 10, 20 and 30, and for each of these values we perform tests on 10, 10 and 15 instances, respectively.
The separation distance, d, that has to be ensured between each pair of aircraft, is set to the standard value 5 NM.The values of the bounds on the variables in equations ( 26 cally, we consider the so-called subliminal speed control that limits speed deviations between −6% and +3% of the original speeds, and heading angle deviations between −π/6 and +π/6.We set the quadrant-penalty parameter to β = 3 (corresponding to equallyspreaded breakpoints for h X , as mentioned above and illustrated on Figure 3.) To handle the issue of potential local infeasibilities mentioned above, we implement a simple multistart heuristic, from different (randomly-generated) starting guesses.
The optimization model is implemented in AMPL [13] and the instances are solved using the IPOPT solver for nonlinear optimization problems [17], version 3.12, with the feasibility tolerance set to 10 −5 .A 2.66 GHz Intel Xeon (octo core) processor with 32 GB of RAM under Linux is used to perform our numerical tests.
Conducting computational tests on the 35 instances yields the following observations.
First, a solution satisfying the separation constraints (zero-value penalty function) is always obtained.At most 2 runs of the multistart heuristic are needed, and this happens only for 2 instances out of the 35 instances studied (one with n = 20 and one with n = 30).All the computed solutions are globally optimal with respect to the penalty objective function (29), as its lower bound zero is attained.
Second, the CPU times (reported in Table 1) reveal that we benefit from the efficiency of continuous optimization methods to compute solutions satisfying the difficult separation constraints, this computation being generally demanding for large-scale instances when such logical constraints are addressed by combinatorial optimization approaches.
Computing times do not exceed 3.09 seconds for instances involving n = 20 aircraft sharing the same airspace, and this number of aircraft is already larger than what is commonly observed in practice when air traffic controllers deconflict cruise flights.Even for the larger instances involving n = 30 aircraft (requiring 3.68 seconds, on average), our tests tend to show that the proposed approach can be useful in the real-world application context considered.The maximum computing time (20.06 seconds) results from the sum of the CPU times of two runs required by the multistart heuristic.The 14 remaining instances involving n = 30 aircraft do not require multistart, and solutions are

Conclusion
Logical constraints are ubiquitous in practical optimization.This paper proposes a continuous-optimization alternative to the two classical, big-M and complementary, mixed-integer formulations of logical constraints: the continuous quadrant penalty formulation.We emphasize some theoretical properties of the 2D smooth non-convex piecewise-quadratic penalty function upon which is based the new approach.We demonstrate the usefulness of our approach on a real-world application, the aircraft conflict avoidance problem, which is commonly addressed by combinatorial optimization, requiring as many extra binary variables as there are possible pairs of vehicles with the classical ways of reformulating logical separation constraints.
This work is in the spirit of the questioning of Conn as to whether penalty functions can have a significant impact in discrete or global optimization.Future work will address other problems involving logical constraints that would incur too numerous extra binary variables.Addressing optimization problems will involve managing a weighting penalty parameter to take into account the original problem's objective.

Figure 1 :
Figure 1: Values of the unique continuous three-piece piecewise-linear Sviolation dipstick function g α (left) and its gradients (right).

Figure 5 :
Figure 5: Level curves of the functions g α and d R 2 \S (which are identical, in black), and those of g β (blue) in the case where α = 1 and β = 3.

Table 1 :
)-(28) are motivated by operational constraints.Specifi-Mean value, standard deviation, minimum and maximum values of the total CPU times (in seconds), for the instances involving n aircraft