Penalty, Barrier and Augmented Lagrangian Methods

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1 Penalty, Barrier and Augmented Lagrangian Methods Jesús Omar Ocegueda González Abstract Infeasible-Interior-Point methods shown in previous homeworks are well behaved when the number of constraints are small and the dimension of the energy function domain is also small. This fact is easily seen, since each iteration of such methods requires of solving a linear equation system whose size depends precisely on the number of constraints and the dimension of the search space. In addition, the energy functions that we could optimize with the previous approaches are restricted to be linear with linear constraints. In this homework I describe three new methods that deal with these inconvenients. I. INTRODUCTION THE, main idea of the kind of methods described here is to construct a sequence of unconstrained optimization problems whose approximated solutions converge to a solution of the original constrained problem. Such a sequence penalizes every non-feasible point. The sequence starts with a small penalty term and this is sequencially inreased with time. At each iteration k we find an approximation x k to the solution of the modified unconstrained problem within a given tolerance τ k. We expect that the sequence {x k } converge to a solution to the original problem, i.e. lim k x k = x. II. THE PROBLEM In order to test the performance of the three methods described here, I will use the energy function given by Uf) = subject to f r g r δ <r,s> C 2 f r f s ) 2 these constraints are equivalent to f r g r + δ 0 g r f r + δ 0 in this context, r, s S, where S is the set of sites in which the function f and g are defined. C 2 is the set of cliques of order 2 and δ is a fixed positive constant. Intuitively, we are looking for the smoothest f for which the constraints are held. The expected result is an edge-preserving regularization of the observed image g. III. THE QUADRATIC PENALTY METHOD A. Equality-constrained problem We will first examine the case in which we have only equality constraints: min x fx) subject to c i x) = 0. The quadratic penalty method constructs the following unconstrained function: Qx, k ) = fx) + 1 c i x) 2 2 k where k > 0 is the penalty parameter. If k 0 then the infeasibilities are increasingly penalized, forcing the solution to be almost feasible. B. The general constrained optimization problem In the general case, we would like to penalize a point x whenever c i x) < 0 but not when c i x) 0. To achieve this we define the operator [ ] as: a)effect of the [ ] operator. [x] = max{ x, 0}. b)quadratic penalty function. Fig. 1. Quadratic penalty function for inequality constraints for different values of. Using this expression we define the function see figure 4)) Qx, ) = fx) + 1 c i x) [c i x)] ) C. Implementation i I In this case, the unconstrained problem is min Qf, ) = f r f s ) 2 + <r,s> C [fr g r + δ] ) 2 + [g r f r + δ] ) 2). Taking the first derivative of Q respect to f r we have: Q f, ) = 2f r f s ) 1 f r [f r g r + δ] Ig r f r δ) 1 [g r f r + δ] If r g r δ) where N r is the first order neighbourhood of the site r, and I is the step function given by { 0, x 0 Ix) = 1, x > 0

TABLE I NUMERICAL RESULTS OBTAINED WITH THE QUADRATIC PENALTY METHOD USING NORMALIZED IMAGES).

1 20 5 144.788 0.01157 bola, δ = 0.05 20 5 228.742 0.00893 taza, δ = 0.2 20 5 9.185 0.

00481 Since d[x] ) 2 = 2[x] I x)) = 2[x] I x) = 2xI x) dx we have Q f, ) = f r 2f r f s ) + 1 f r

r + Ig r f r δ) If ) r g r δ) = f s + 1 δ g r)ig r f r δ) 1 δ+g r)if r g r δ) Since the value of I

δ g r)i 1 1 δ + g r)i 2 2 N r + I1 I2 D.

original image, the solution found, the estimated lagrange multipliers, the numerical value of the

a)optimal image found. b)λ. c) λ. Fig. 2. Results obtained for δ = 0.2 first row), δ = 0.

Conclusions The quadratic penalty method leads us to a simplification of the constrained

In the case of quadratic energy functions, Gauss-Seidel approach can be used with excelent

function, then we can not know how good is the result obtained.

In the case of image processing, a slight violation of the constraints is not important but in

2 TABLE I NUMERICAL RESULTS OBTAINED WITH THE QUADRATIC PENALTY METHOD USING NORMALIZED IMAGES). Iter. Optimum value Mean constraint violation bola, δ = bola, δ = bola, δ = taza, δ = taza, δ = taza, δ = Since d[x] ) 2 = 2[x] I x)) = 2[x] I x) = 2xI x) dx we have Q f, ) = f r 2f r f s ) + 1 f r g r + δ)ig r f r δ) + 1 g r f r + δ)if r g r δ) then, making Q f r f, ) = 0 we obtain: = 2 f r 2 N r + Ig r f r δ) If ) r g r δ) = f s + 1 δ g r)ig r f r δ) 1 δ+g r)if r g r δ) Since the value of I is not in terms of f r we can use the Gauss-Sidel iterative scheme given by where f r = 2 f s + 1 δ g r)i 1 1 δ + g r)i 2 2 N r + I1 I2 D. Experimental results I 1 = Ig r f r δ) I 2 = If r g r δ) For the experiments below I show the original image, the solution found, the estimated lagrange multipliers, the numerical value of the energy function, the average of the violation of the constraints v defined as v = 1 S [c i x )] a)optimal image found. b)λ. c) λ. Fig. 2. Results obtained for δ = 0.2 first row), δ = 0.1 second row) and δ = 0.05 third row). E. Conclusions The quadratic penalty method leads us to a simplification of the constrained optimization problem which can be solved using conventional methods. In the case of quadratic energy functions, Gauss-Seidel approach can be used with excelent results, at least visually; it is difficult even to know what is the optimal value of the energy function, then we can not know how good is the result obtained. The convergence is fast and the results are good, but the constraints are slightly violated. In the case of image processing, a slight violation of the constraints is not important but in cases in which it is so, a correction must be done to x to make it feasible. Finally, the results on noisy images are not as good as on non-perturbed images. IV. LOGARITHMIC BARRIER METHOD As we can see, the quadratic penalty method has the inconvenient of slightly violating the constraints. We saw that in the case of image processing this is not too important. Barrier

3 Our problem becomes min P f, k ) = <r,s> C 2 f r f s ) 2 k [logf r g r + δ) + logg r f r + δ)] Fig. 3. a)noisy image, σ = 0.1. b)result with δ = Results obtained by the quadratic penalty method on noisy images. methods are used in cases in which it is very important the solution to be feasible, the general formulation of these methods begins by defining a Barrier function, which is a function T defined only in F 0 for which the following propierties hold: lim x x0 T x) = x 0 F 0 T is smooth into F 0 where F 0 is the strictly feasible set: F 0 = {x R n c i x) > 0 i I} The most important barrier function is the logarithmic barrier function defined as: T x) = i I logc i x)). The unconstrained optimization problem is, then, minimize P x, ) = fx) i I logc i x)). where is the barrier parameter that in is fact written as k and lim k k = 0.As we can see, it is necesary to have a feasible starting point and, clearly, it is necesary F 0 to be non-empty. The presence of the log function makes it harder to minimize this function than the quadratic penalty function. A. Implementation Once again, we will try to solve the problem described in section II, using the log-barrier method. We can make use of any of the conventional strategies for unconstrained optimization, taking care of avoid leaving the feasible set { 0. For this homework I implemented a simple gradient descense iterative method. In this case we have P x, ) = fx) i I which leads us to yhe iterative scheme x t+1 = x t h fx t ) i c i x t ) c i x t ) c i x t ) c i x t ) P f, k ) = 2 f r f s ) k [logf r g r + δ) + logg r f r + δ)] = 2 [ ] 1 f r f s ) k f r g r + δ) 1 g r f r + δ) The modification we have to do to avoid leaving the strictly feasible set is the following: Let φ t r = h P r f t, t ), we will leave the strictly feasible set if either fr t+1 g r + δ 0 fr t + φ t r g r + δ 0 φ t r g r f r δ or g r fr t+1 +δ 0 g r fr t φ t r +δ 0 φ t r g r f r +δ. Let Ω be the subset Ω S of sites for which one of the conditions above holds. Then we choose α = 1 if Ω is empty and if Ω is non-empty α = min r Ω { gr f r δ φ t r if the first condition holds and α = min r Ω { gr f r + δ φ t r if the second condition holds. To ensure the next point is strictly feasible, we must set x t+1 = x t + α1 ɛ)φ t. B. Experimental results This method has many problems to converge. The minimum value of the energy function is greater that the obtained with the quadratic penaty method. Visually, the regularization is notably worst than the obtained with the previous method. I will show the results in the same cases except on the noisy images. I think that, given the results, we can not expect good news on these. The Lagrange multipliers are difficult to estimate because the constraints quickly take values close to zero. } }

4 this is the Lagrangian with a quadratic penalty term on the constraints. Fig. 4. a)δ = 0.2 b)δ = 0.1 c)δ = 0.05 Results using log-barrier method. TABLE II NUMERICAL RESULTS OBTAINED WITH THE log-barrier METHOD USING NORMALIZED IMAGES). Iter. Optimum value Mean constraint violation bola, δ = bola, δ = bola, δ = taza, δ = taza, δ = taza, δ = C. Conclusions Given the experimetal results, it is clear that the gradient descent method used is not a good choice, the optimal values found are very poor either numericaly and visually. This method rapidly becomes unstable, and we need to monitor de numerical value of the variables to avoid them to become zero or infinity. The constraints violation is clearly smaller than the obtained with quadratic penalty, and in fact this is due to numerical error, because the form in which we avoid leaving the strictly feasible set is teoretically correct. V. AUGMENTED LAGRANGIAN METHOD When we use the quadratic penalty method, the equality constraints are not satisfied. Instead, the value of these are c i x k ) = k λ i i E instead, k 0, c i x) 0, making the constraint to be satisfied. The lagrangian method avoid having this infeasibilities through the estimation of the lagrange multipliers. The Augmented Lagrangian Function is defined as L A x, λ, ) = fx) λ i c i x) + 1 c i x) 2 2 Taking the first derivative of this function, we have L A x, λ, ) = fx) λ i c i x) + 1 c i x) c i x) = fx) c i x) λ i c ) ix) In an iterative scheme, we can see from the previous expression that λ i λ k i c ix k ) ) k when x k x. This property, motivates the Method of Multipliers-Equality Constraints, that consists on the same scheme as before, generating a sequence of partial solutions {x k } minimizing the Augemnted Lagrangian function leaving λ k fixed and at each iteration we set λ k+1 i = λ k i c ix k ) ) k Now, we need an extension of this idea for inequality constraints. This is seen in the next subsection. A. Augmented Lagrangian for inequality constraints The technique we will use to handle inequality constraints is the same we used when we studied the Simplex method, this is, we will introduce slack variables. First, asume that we have just inequality constraints. Given the problem we reformulate it as min fx) subject to c i x) 0 i I min fx) subject to c i x) s i = 0 i I s i 0. This expression seems to be helpless due to the presence of the new constraints s i 0, i I. The difference is that now the inequality constraints are linear and, as we will see, it is easier to handle this constraints. By writing the Augmented Lagrangian Function, our new problem is min L A x, λ, ) = fx) λ i c i x) s i )+ 1 c i x) s i ) 2 2 subject to s i 0. Now lets see which values of s i minimize this function L A s i x, λ, ) = λ i 1 c ix) s i )

5 this function has a critical point in s i = c i x) λ i since the function is quadratic with respect to s i we have that the restricted minimizer is s i = max{c x) λ i, 0}. Using this expression, and substituting its value on the original problem, we see that λ k i c i x) s i ) c ix) s i ) 2 = λ k i c ix) c ix) 2 if c i x) λ k i 0 2 λk i )2 otherwise In order to make this expression to be clear, we introduce the function Ψ given by Ψt, σ, ) = σt t2 if t σ 0 2 σ2 otherwise Finally, we obtained the transformed problem min L A x, λ k, k ) = fx) + i I Ψc i x), λ k i, k ) if we compare the derivative of this function = fx k ) x L A x k, λ k, k ) i I c ix k ) λ k i λ k i c ix k ) ) c i x k ) 0 k with the first KKT condition for an optimal point fx ) i I c ix )=0 λ i c i x ) = 0 we see that the values of the Lagrange multipliers should be λ i λ k i c ix k ) k then keeping their values to be nonnegative we know that the Lagrange multipliers must be nonnegative) we can construct the sequence of partial solutions {x k } as before but with the extra step of actualizing the values of the lagrange multipliers using the formula λ k+1 i = max{λ k i c ix k ) k, 0} TABLE III NUMERICAL RESULTS OBTAINED WITH THE AUGMENTED LAGRANGIAN METHOD USING NORMALIZED IMAGES). Iter. Optimum value Mean constraint violation bola, δ = bola, δ = bola, δ = taza, δ = taza, δ = taza, δ = B. Implementation For the problem stated in II, the Augmented Lagrangian for inequality constraints has the form L A f, λ, λ, ) = = f r f s ) 2 + Ψc r f), λ r, )+Ψ c r f), λ r, )) <r,s> C 2 where c r f) = g r f r + δ c r f) = f r g r + δ taking the first derivative with respect to f r we have L A f, λ, f λ, ) = 2 f r f s ) r [ + λ r g ] [ r f r + δ) I 1 + λ r + f ] r g r + δ) I 2 where I is the step function defined above and I 1 = Iλ r g r f r + δ)) I 2 = I λ r f r g r + δ)) Again, the expression for I is not in terms of f r, then we can apply the Gauss-Seidel iterative scheme to minimize the Augmented Lagrangian 2 [ ] f s + gr+δ) λ r I 1 + [ λr δ gr)] I 2 f r = 2 N r + I1 + I2 C. Experimental results In fugure 5) I show the optimal images found applying the Augmented Lagrangian Method on the same images as before. The quality of the images seems to be equal than the obtained with the quadratic penalty method, but by evaluating the energy function of the result we can see that the Augmented Lagrangian Method si slightly better see table III). The main difference is seen on the lagrangian multipliers. As we said above, the problem can be seen as an Edge Preserving Regularization, then it is expected the Lagrange multipliers to be considerably biger in the edges of the image than in non-endge sites. This property can not be seen on the images given by the quadratic penalty method, but the images obtained

1 rows 2 & 5) and δ = 0.05 rows 3 & 6). using the Augmented Lagrangian show clearly this property.

6 a)noisy image, σ = 0.1. b)result with δ = 0.2. Fig. 6. Results obtained by the Augmented Lagrangian method on noisy images. results for the test on noisy images are similar than the obtained eith the quadratic penalty method. a)result. b)λ. c) λ. d)λ + λ. Fig. 5. Results obtained for δ = 0.2 rows 1 & 4), δ = 0.1 rows 2 & 5) and δ = 0.05 rows 3 & 6). using the Augmented Lagrangian show clearly this property. Clearly, since an edge is defined by two sites say < r, s >, if the lagrange multiplier corresponding to the inequality c r f) is activated then the multiplier corresponding to the inequality c s f) should be activated on s. This property can also be clearly seen on the image formed by the sum of the Lagrange multipliers the edges are wider than any of the other two images). Another important characteristic that can be seen in table III is that the mean constraint violation is considerably smaller than the obtained with the quadratic penalty method. D. Conclusion Given the results, both numerical and visual), and the characteristics mentioned in the previous section, there is no much to say... This method is considerably better than the other two methods developed in this homework. In fact this method improves all the defitiencies presented by its competitors. The

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