Notice that iii is just the original constraint. The global optimum can be found by comparing the values of the original objective function at the points satisfying the necessary and locally sufficient conditions. On 9 April the guards at Milag-Marlag moved out and were replaced by older men, presumably local Volkssturm.
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Often the Lagrange multipliers have an interpretation as some quantity of interest. For example, if the Lagrangian expression is. As examples, in Lagrangian mechanics the equations of motion are derived by finding stationary points of the action , the time integral of the difference between kinetic and potential energy. In control theory this is formulated instead as costate equations. Moreover, by the envelope theorem the optimal value of a Lagrange multiplier has an interpretation as the marginal effect of the corresponding constraint constant upon the optimal attainable value of the original objective function: For example, in economics the optimal profit to a player is calculated subject to a constrained space of actions, where a Lagrange multiplier is the change in the optimal value of the objective function profit due to the relaxation of a given constraint e.
Sufficient conditions for a constrained local maximum or minimum can be stated in terms of a sequence of principal minors determinants of upper-left-justified sub-matrices of the bordered Hessian matrix of second derivatives of the Lagrangian expression.
Evaluating the objective function f at these points yields. In this example we will deal with some more strenuous calculations, but it is still a single constraint problem.
Notice that iii is just the original constraint. In other words, we wish to maximize the Shannon entropy equation:. Carrying out the differentiation of these n equations, we get. By using the constraint. Hence, the uniform distribution is the distribution with the greatest entropy, among distributions on n points.
The critical points of Lagrangians occur at saddle points , rather than at local maxima or minima. For this reason, one must either modify the formulation to ensure that it's a minimization problem for example, by extremizing the square of the gradient of the Lagrangian as below , or else use an optimization technique that finds stationary points such as Newton's method without an extremum seeking line search and not necessarily extrema.
This problem is somewhat pathological because there are only two values that satisfy this constraint, but it is useful for illustration purposes because the corresponding unconstrained function can be visualized in three dimensions. Using Lagrange multipliers, this problem can be converted into an unconstrained optimization problem:. In order to solve this problem with a numerical optimization technique, we must first transform this problem such that the critical points occur at local minima.
This is done by computing the magnitude of the gradient of the unconstrained optimization problem. There was a camp theatre in Marlag and the POWs performed concerts and plays. Each camp had its own sports field and there was also a library with around 3, books.
Prisoners ran courses in languages and mathematics, as well as commercial, vocational, economic and scientific subjects. POWs were allowed to send two letters and four postcards each month. There were no restrictions on the number of letters a POW could receive. Naturally all incoming and outgoing mail was censored. Under normal conditions the camps had a capacity of 5, According to official figures in April there were 4, men held there.
Initially the camp was guarded by Naval troops. Later they were replaced by Army reservists. The German Navy also operated a Dulag Du rchgangs lag er , "Transit camp" in Wilhelmshaven , where newly arrived prisoners were processed before being sent to other camps. After the Allied bombing raids on Wilhelmshaven in February this facility was moved to Westertimke. The camp Dulag Nord was located between Marlag and Milag. To the north and east of the village three smaller camps were also built.
The Kommandatur contained the headquarters and administration buildings, while the Stabslager and the Wache contained accommodation for the administrative personnel and the camp guards. At the end of prisoners evacuated from other camps began to arrive, resulting in overcrowding, and a reduction in food rations. On 2 April the Commandant announced that he had received orders to leave the camp with most of his guards, leaving only a small detachment behind to hand over the camp to Allied forces, who were already in Bremen.
However that afternoon a detachment of over a hundred SS-Feldgendarmerie entered the camp, mustered over 3, men and marched them out, heading east. The next day, at around at Over the next few days the column was attacked from the air several times. On 9 April the guards at Milag-Marlag moved out and were replaced by older men, presumably local Volkssturm. On 19 April units of the 15th Panzergrenadier Division positioned tanks and artillery next to the camps. The remaining prisoners responded to the threat of a pitched battle on their doorstep by digging slit trenches.
The artillery fired from the positions next to the camps,  but fortunately had moved away by the time the British Guards Armoured Division liberated the camps on 27 April