«End-to-End QoS Provision over Heterogeneous IP and non IP Broadband Wired and Wireless Network Environments A dissertation submitted in satisfaction ...»
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A.1 Regional and Functional Constraints Throughout this report we have considered optimization problems that were subject to constraints. These include the problem of allocating a ﬁnite amounts of bandwidth to maximize total user beneﬁt the social welfare maximization problem, and the time of day pricing problem. We make frequent use of the Langragian method to solve these problem. This appendx provides a tutorial on the method.
Take for example:
Here f is to be maximized subject to constraints that are of two types. The constraint x ∈ X is a regional constraint. For example, it might be x 0. The constraint g(x) = b is a functional constraint Sometimes the functional constraint is an inequeality constraint, like g(x0) b But if it is, we can always add a slack variable z, and re-write it as the equality constraint g(x) + z = b, re-deﬁning the regional constraint as x ∈ X and z geqslant0. To illustrate, we shall use the
NETWORK problem with just on resource constraint:
where b is a positive number.
A.2 The Langragian Method The solution of a constrained optimization problem can often be found by using
the so-called Langragian method. We deﬁne the Langragian as:
In general, the Langragian is the sum of the orignal objective function, and a term thal involves the functional constraint add a Langrage multiplier λ. Suppose we ignore the functional constraint and consider the problem of maximizing the Langragina, subject only to the regional constraint. This is often an easier problem than the original one. The value of x that maximizes L(x, λ) depends upon the value of λ. Ket us denote this optimizing value of x by x(λ).
For example, since L1 (x, λ) is a concave function of x, t has a unique maximum
at a point where f is stationary with respect to changes in x, i.e. where:
Suppose there existe x∗ ∈ X and λ∗, such that x∗ maximized L(x, λ∗ ), and g(x∗ ) = b. Then x∗ solves P.
Equality in the ﬁrst line holds because we have simply added 0 on the right hand side. The inequaliy in the secodn line hoolds because we have enlarged the set over which maximization takes place. In the third line, we use the fact that x∗ maximizes L(x, λ∗ and in the fourth line we use g(x∗ = b. But x∗ is feasible for P, in that it satisﬁes the regional and functional constraints. Hence x∗ is optimal.
If g and b are vectors, so that g(x) = b expresses more than one constraint, when we would write
B.1 The case of producers and consumers
In this appendix we prove that under the tatonnement mechanism price convergence. Consider ﬁrst the problem of maximizing social surplus:
A typical instance of this problem is when xr is the ﬂow in route r through the network generateing value ur (xr ). A route is a set of links; Ajr = 1 if route r uses link j. Cj is the capacity if link j.
Since C − Ax is the vector excess demand and µ is the price vector, we can repeat the last steps in previous section, and show that again in this case tatonnement will converge to the socially optimal vector of prices.