## When does strong duality fail in linear programming? - Computer Science Stack Exchange

Network Structure. The following are the various types of organization structure:. The following analysis gives the nature, merits and demerits of these structures:. It is also known as Scalar organization or Military form of organization.

It is the oldest form of organization. Under this method, authority flows in a vertical manner form top to bottom. Each position in the structure has an authority over a lower position. Line organization is of two types, viz. Under pure line organization all perform same type of work at any level. On the other hand, under departmental line organization, each department performs different type of work.

Each individual knows clearly to whom he is responsible. It ensures personal contact between workers and manager. It ensures discipline between the employees and employer. There is concentration of authority at the top level.

It leads to autocratic control. Communication flows from top to bottom. There is no participation the comparison contrast essay workers. It lacks specialization as a line manager is responsible for both planning and execution of work.

Overburdens the top executives and they cannot perform their work properly. In spite of these limitations, it is suitable for small concerns where there are fewer levels of authority. This concept was devised by F. It is based on the functional foremanship. Under this method, the structure of the enterprise is classified into different functional areas.

Each functional area is headed by a specialist who has full control of that function over the organization and gives instructions direct to the personnel, rather than through the chain of command. Each chief manager is in charge of a particular activity. For example, chief finance manager is in charge of finance function of all the three units. Thus, he enjoys functional authority over the employees of other departments also. Subordinates can make use of the expertise of functional experts.

The burden of top executive is reduced as each expert looks after only one function. There is expert and better control can be exercised by functional experts.

It provides better scope for expansion and diversification. It is against the principle of unity of command as the subordinate is responsible to a number of superiors. Lower level people are line and staff organization given linear programming duality opportunity for all round experience and hence they do not fit into top level positions.

The complex nature of functional organization with its cross relationship creates a confusion among workers. The decision making is very slow as involvement of many experts is required. There is a scope of lack of coordination as each specialist thinks of only his function and ignores other functions. It is suitable to all kinds of organization provided applied at higher levels. At lower levels too many cross relationship creates confusion throughout the organization. Line and staff organization is a combination of functional and line structure.

Line authority flows from top to bottom and the line executive is directly concerned with the accomplishment of primary objectives. They are actual doers and generally do not possess specialized knowledge to solve complex problems. To provide specialized assistance to line mangers, staff positions are created. Staff means a stick in the hand for support. Thus, staff helps the line executives in their work. They play the role of an advisor.

The other approach views that line and staff are two kinds of line and staff organization. According to this approach, line authority is defined as a direct authority which a superior exercises over his subordinates to carry out orders and instructions. The exercise of this authority is always downwards, that is, form a superior to a subordinate. Staff authority involves giving advice to line managers to carry on the operation. The flow of this authority may be in any direction depending on the need of such an advice.

It is common that in actual practice, some variations may exist. The variations are more pronounced in the case of staff authority. The distinction between line and staff though not rigid, is important because staff must be provided if the growing organization is to accomplish its goals.

The differentiation between line and staff is necessary for the following reasons. In line and staff organization, the line authority remains the same in the organization. But staff executives are attached with line executives who help them by providing necessary advice on important matters.

# Category extension vs line extension

Staff executives have no power to command subordinates in other departments. In most business units, staff executives are used for collecting data required for taking decisions and to provide expert advice to line managers. This combination provides for specialized knowledge where staff executives guide and advise line executives. It reduces the burden of line executives because staff carries on detailed investigation of each activity. Better decisions are possible as expertise of the staff is used.

It is flexible in the sense that staff can be added to line executives. Always it creates confusion as it is very difficult to define the authority relationship between line and staff. Staff executives are not accountable and they may not take the tasks seriously. There is always constant conflict between line and staff because the nature of their functions differs from each other.

Line and staff relationship implies that both are complimentary in nature. However, there are frequent instances of conflict between line and staff in the organizations, resulting in friction.

The various factor leading to line-staff conflict can be grouped into three categories as follows:. Line executives are the actual doers who responsible for the accomplishment of goals.

From their view point staff executives create conflicts in the following ways:. Thread Modes What are the features of line and staff organisation. Find Reply nbm Posting Freak Posts: 3, Threads: Joined: Sep 2AM Line and staff organisation refers to a pattern in which staff specialists advise line managers to perform their duties. Find Reply. What is Line organisation, its features, advantages and limitations?

Does Personnel Management do both Line and staff function. Difference between Project Organisation and Matrix Organisation. Simple Soul. Kanwal Ishfaq. Azliana Mohd Taib. Errol Lucero. Lainnya Dari Narayana Reddy. Narayana Reddy. Session a white heron essay Environmental Appraisal and Organizational Appraisal. Populer di Human Activities.

Jung Jung. Richard Saxton. Make your procedures beautiful and engaging, on any device. As people read a process - grab ideas to improve it. Embed videos and screencasts to make everything far more engaging. Have co-workers value and engage with your operations manual. Save countless, painful hours formatting every document by hand. Go from static documentation to live, actionable processes - instantly.

By making ourselves write down our processes and know-how on Tallyfy - we can now ensure that steps are never missed or done out of order. This is the most traditional of the organizational structures that businesses use. The advantage of this type of organizational structure lies in its simplicity. The disadvantage lies in its rigidity and the length of time needed for information to flow through the organization.

What are Alternatives montaigne some lines virgil Layoffs? When a company is making large losses, laying off employees can be an attractive option for cutting costs. In addition to wages, the organization will save on health care insurance, office space, overheads and a host of other expenses.

But when aAs the dual problem has lesser number of constraints than the primal 2 instead of 4it requires lesser work and effort to solve it. As you move, you're wandering a little away from precision, from perfection, but if you don't take too big a step, Newton is safe.

Maybe since this is a course in scientific computing, I should've written on the very first category extension vs line extension in big letters Newton, because that idea of following the gradient is the central method of solving non-linear equations. And then on the board beneath I would have written in big letters Carefully because the derivative is a local thing.

## Linear programming

And if you follow the derivative a long distance, follow the derivative here a long distance out to here, who knows what -- you've lost the safety of Newton's method. So Newton's method always comes in reality with some kind of a trust region, some region where you can rely on the derivative being a reasonable approximation of the way the function is moving.

OK, so we do that here too. What does Newton's method do actually? So Newton's method we're at a particular x, y, s, and we've got to move. So the unknowns are the components of xthe components of y, and the components of s.

So Newton's method takes steps: a delta x, a delta y, and a delta s, computes what those should be, and then that gives the direction, and if you take them exactly, that's the full Newton set, which you would be very happy to do because that gives terrific conversion, but if it's too big a step, then you have to cut back. So the equations for these are what you need so there'll be an A delta x will be 0. Because turabian line spacing isn't changing.

There will be a A transposed delta y, a transposed delta y plus delta s will be 0 because the c category extension vs line extension changing. And then we'll get an equation out of this, which is a really significant one, but maybe time is running out on and I'm not going to do justice to. But that's a non-linear term, where you see if I keep A delta x 0, then my new x is exactly feasible right? If I'm at an Ax equals b and I move it by a delta x that's in the null space, then I have -- All I'm saying is that when I take that step I will have A x plus delta x still equal to b.

Constraints still satisfied. When I take this step, since that's linear, the constraint when I add on the delta y and the delta s and the 0 I still have -- my new point still satisfies that constraint. But this is of course not exactly satisfied. If I had the solution to this, I'd be done. That's my problem. Anyway, so it's statistics about homework exactly satisfied.

Newton would tell you a linearization of it, and linear programming duality would move in that gradient direction to try to make the things -- to try to make equality hold, because our current x doesn't have equality hold. And of course the c is a transposed y plus s. So that equation -- You see what's going on here? This is a transposed y plus s, and the x is multiplying those, so there's a product there. And when I take the derivative it's a product rule. I get two terms.

Anyway, I get a third equation from here for delta x that connects delta x, delta y, and delta s. I take that step and that's my interior point method. That's my Newton step. So maybe I just end by reporting results, so I'll end with just two comments.

First is, is the method any good? And of course you only know by trying. And the answer is yeah. Typically you get the gap -- you get the duality gap down below 10 to the minus 8, which is usually very satisfactory in 20 to 80 steps. You can never prove a statement like that because you can always create some awful example, but this is the typical performance of the method.

Which is pretty good, regardless of m and n. That's what's wonderful -- that the number of steps doesn't increase with the size of the problem.

Of course, the cost per step does increase with the size of the problem. OK so that's the results, and that's why the method is popular. And now I just wanted to not leave duality, which is such a key idea, without going back to our much more familiar problem of quadratics, where there are quadratic terms.

And the best model you remember was projection.

## Montaigne some lines virgil

You remember that we had a victor b and we have the line, the null space of A. This was the column space of A. This was all Ax's.

And perpendicular to it was the null space of A transpose. All A transpose y should equal 0. Do you remember this? This was the model problem for understanding. So that the projection of this solved one problem. The projection in the other direction -- we called that p. This was the projection p equal A times the best x.

The projection in the opposite direction found the e, the error, but it was the solution to the dual problem. And now I want to say where was duality in this picture? Well duality was -- let me call it e hat, the winning, the projection, the right guy over here.

Or maybe y hat. OK where was duality? Montaigne some lines virgil came in this case in the fact that it was Pythagoras. Duality in this simple, beautiful problem was simply the fact that p squared, this winner squared, plus e squared, was b squared. The winners were the orthogonal projection.

# Complementary Slackness Theorem (Duality Theory in Linear Programming)

And now where is weak duality? It's the last second of the lecture. Weak duality says take something that's allowed, like that and take something that's allowed here, like that. And then those are not the winner. Get Started.

Top Online Courses Chevron Down. Top Online Specializations Chevron Down. Online Certificates Chevron Down. In particular, one very interesting thing to do is to observe what happens in the dual when we apply the simplex method or the interior point method to the primal linear program.

For the simplex method, this naturally defines a new method called the dual simplex method. Learn more by reading my article on simplex methods. An important application line and staff organization the duality theory is the definition of the prices turabian line spacing the ressources. As a matter of fact, this algorithm consists in minimizing the total cost of production of electricity with constraints on the demand at each geographical location.

The demand can be viewed as ressource. If the demand at a location increases, then it would cause an incremented cost of production of electricity. In the electricity market, this incremented cost is considered as the price of electricity at that point.

If a demand is at linear programming duality location far from generators that induce plenty of losses, an increase of the demand at that location will induce an important incremental production, hence a higher incremental cost of production. Therefore, the dual variable associated to the demand, which is the price, will be higher.

An intuitive approach is given. But that's not all. We present an important variant called the dual simplex. Finally, we'll explain its main default, that is, when facing degeneracy.

Half of the time, it's what's used to solve real-world problems! My PhD supervisor effectively took great advantage of these tricks and founded companies with it. This article explains the tricks. The dual variable stabilization is a very recent state-of-the-art way to deal with degeneracy. An intuitive understanding is given in this article. Now I have a profound understanding of what Dual does. Thank you very much. Your email address will not be published.

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Find out what you can do. The duality theorem has a physical interpretation too. From Wikipedia, the free encyclopedia. The dual of a given linear program LP is another LP that is derived from the original the primal LP in the following schematic way: Each variable in the primal LP becomes a constraint in the dual LP; Each constraint in the primal LP becomes a variable in the dual LP; The objective direction is inversed - maximum in the primal becomes dissertation romantisme exemple in the dual and vice-versa.

Understanding and Using Linear Programming. Berlin: Springer. Pages Ahmadi Princeton University. Categories : Linear programming. Namespaces Article Talk. Views Read Edit View history. Evolutionary algorithm Hill climbing Local search Simulated annealing Tabu search. Complementarity problems and algorithms. Simplex Dantzig Revised simplex Criss-cross Lemke.

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Covering problems. Packing problems. Minimum set cover. Maximum set packing. Maximum independent set. Unsolved problem in computer science : Does linear programming admit a strongly polynomial-time algorithm?

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# Turabian line spacing

Basic version available as a standard add-on for Excel. Solver with parallel algorithms for large-scale linear programs, quadratic programs and mixed-integer programs.

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