# 2.1 Constrained optimization  (Page 2/2)

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A typical problem arising in signal processing is to minimize $x^TAx$ subject to the linear constraint $c^Tx=1$ . $A$ is a positive definite, symmetric matrix (a correlation matrix) in mostproblems. Clearly, the minimum of the objective function occurs at $x=0$ , but his solution cannot satisfy the constraint. The constraint $g(x)=c^Tx-1$ is a scalar-valued one; hence the theorem of Lagrange applies as there are no multiple components in theconstraint forcing a check of linear independence. The Lagrangian is $L(x, )=x^TAx+(c^Tx-1)$ Its gradient is $2Ax+c$ with a solution ${x}_{}=-\left(\frac{{}_{}A^{(-1)}c}{2}\right)$ . To find the value of the Lagrange multiplier, this solution must satisfy the constraint. Imposing theconstraint, ${}_{}c^TA^{(-1)}c=-2$ ; thus, ${}_{}=\frac{-2}{c^TA^{(-1)}c}$ and the total solution is ${x}_{}=\frac{A^{(-1)}c}{c^TA^{(-1)}c}$

When the independent variable is complex-valued, the Lagrange multiplier technique can be used if care is taken to make the Lagrangian real. If it is not real, wecannot use the theorem that permits computation of stationary points by computing the gradient with respect to $\overline{z}$ alone. The Lagrangian may not be real-valued even when the constraint is real. Once insured real, the gradientof the Lagrangian with respect to the conjugate of the independent vector can be evaluated and the minimizationprocedure remains as before.

Consider slight variations to the previous example: let the vector $z$ be complex so that the objective function is $(z)Az$ where $A$ is a positive definite, Hermitian matrix and let the constraint be linear, but vector-valued ( $Cz=c$ ). The Lagrangian is formed from the objective function and the real part of the usual constraint term. $L(z, )=(z)Az+()(Cz-c)+^T(\overline{C}\overline{z}-\overline{c})$ For the Lagrange multiplier theorem to hold, the gradients of each component of the constraint must be linearlyindependent. As these gradients are the columns of $C$ , their mutual linear independence means that each constraint vector mustnot be expressible as a linear combination of the others. We shall assume this portion of the problem statement true.Evaluating the gradient with respect to $\overline{z}$ , keeping $z$ a constant, and setting the result equal to zero yields $A{z}_{}+(C){}_{}=0$ The solution is ${z}_{}$ is $-(A^{(-1)}(C){}_{})$ . Applying the constraint, we find that $CA^{(-1)}(C){}_{}=-c$ . Solving for the Lagrange multiplier and substituting the result into the solution, we find that thesolution to the constrained optimization problem is ${z}_{}=A^{(-1)}(C)CA^{(-1)}(C)^{(-1)}c$ The indicated matrix inverses always exist: $A$ is assumed invertible and $CA^{(-1)}(C)$ is invertible because of the linear independence of the constraints.

## Inequality constraints

When some of the constraints are inequalities, the Lagrange multiplier technique can be used, but the solution must bechecked carefully in its details. But first, the optimizationproblem with equality and inequality constraints is formulated as $\min\{x , f(x)\}\text{subject to}g(x)=0\text{and}h(x)\le 0$ As before, $f()$ is the scalar-valued objective function and $g()$ is the equality constraint function; $h()$ is the inequality constraint function .

The key result which can be used to find the analytic solution to this problem is to first form the Lagrangian in the usualway as $L(x, , )=f(x)+^Tg(x)+^Th(x)$ . The following theorem is the general statement of the Lagrange multiplier technique for constrained optimizationproblems.

Let ${x}^{}$ be a local minimum for the constrained optimization problem. If the gradients of $g$ 's components and the gradients of those components of $h()$ for which ${h}_{i}({x}^{})=0$ are linearly independent, then $\frac{d L({x}^{}, {}^{}, {}^{})}{d x}}=0$ where ${}^{}\ge 0$ and ${}_{i}^{}{h}_{i}({x}^{})=0$

The portion of this result dealing with the inequality constraint differs substantially from that concerned with theequality constraint. Either a component of the constraint equals its maximum value (zero in this case) and thecorresponding component of its Lagrange multiplier is non-negative (and is usually positive) or a component is less than the constraint and its component of the Lagrange multiplier is zero. This latter result meansthat some components of the inequality constraint are not as stringent as others and these lax ones do not affect thesolution.

The rationale behind this theorem is a technique for converting the inequality constraint into an equalityconstraint: ${h}_{i}(x)\le 0$ is equivalent to ${h}_{i}(x)+{s}_{i}^{2}=0$ . Since the new term, called a slack variable , is non-negative, the constraint must be non-positive. With the inclusion of slack variables, theequality constraint theorem can be used and the above theorem results. To prove the theorem, not only does the gradientwith respect to $x$ need to be considered, but also with respect to the vector $s$ of slack variables. The ${i}^{\mathrm{th}}$ component of the gradient of the Lagrangian with respect to $s$ at the stationary point is $2{}_{i}^{}{s}_{i}^{}=0$ . If in solving the optimization problem, ${s}_{i}^{}=0$ , the inequality constraint was in reality an equality constraint and that component of the constraintbehaves accordingly. As ${s}_{i}=\sqrt{-{h}_{i}(x)}$ , ${s}_{i}=0$ implies that that component of the inequality constraint must equal zero. On the other hand, if ${s}_{i}\neq 0$ , the corresponding Lagrange multiplier must be zero.

Consider the problem of minimizing a quadratic form subject to a linear equality constraint and an inequality constrainton the norm of the linear constraint vector's variation. $\min\{x , x^TAx\}\text{subject to}(c+)^Tx=1\text{and}()^{2}\le$ This kind of problem arises in robust estimation. One seeks a solution where one of the "knowns" of the problem, $c$ in this case, is, in reality, only approximately specified. Theindependent variables are $x$ and  . The Lagrangian for this problem is $L(\{x, \}, , )=x^TAx+((c+)^Tx-1)+(()^{2}-)$ Evaluating the gradients with respect to the independent variables yields $2A{x}^{}+{}^{}(c+{}^{})=0$ ${}^{}{x}^{}+2{}^{}{}^{}=0$ The latter equation is key. Recall that either ${}^{}=0$ or the inequality constraint is satisfied with equality. If ${}^{}$ is zero, that implies that ${x}^{}$ must be zero which will not allow the equality constraint to be satisfied. The inescapable conclusion isthat $({}^{})^{2}=$ and that ${}^{}$ is parallel to ${x}^{}$ : ${}^{}=-(\frac{{}^{}}{2{}^{}}{x}^{})$ . Using the first equation, ${x}^{}$ is found to be ${x}^{}=-\left(\frac{{}^{}}{2}\right)A-\frac{{}^{}^{2}}{4{}^{}}I^{(-1)}c$ Imposing the constraints on this solution results in a pair of equations for the Lagrange multipliers. $(1/4\frac{{}^{}^{2}}{{}^{}})^{2}c^T(A-1/4\frac{{}^{}^{2}}{{}^{}}I)^{-2}c=$ $c^TA-1/4\frac{{}^{}^{2}}{{}^{}}I^{(-1)}c=-\left(\frac{2}{{}^{}}\right)-2\frac{{}^{}}{{}^{}^{2}}$ Multiple solutions are possible and each must be checked. The rather complicated completion of this example is left tothe (numerically oriented) reader.

do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
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