Lagrange multiplier interpretation

The Lagrange multipliers can be interpreted as MO energies, i.e. they expectation value of the Fock operator in the MO basis (multiply eq. (3.41) by the left and integrate). 

The canonical MOs are convenient for the physical interpretation of the Lagrange multipliers. Consider the energy of a system with one electron removed from orbital number Ic, and assume that the MOs are identical for the two systems (eq. (3.32)). 

So far there have not been any restrictions on the MOs used to build the determinantal trial wave function. The Slater determinant has been written in terms of spinorbitals, eq. (3.20), being products of a spatial orbital times a spin function (a or /3). If there are no restrictions on the form of the spatial orbitals, the trial function is an Unrestricted Hartree-Fock (UHF) wave function. The term Different Orbitals for Different Spins (DODS) is also sometimes used. If the interest is in systems with an even number of electrons and a singlet type of wave function (a closed shell system), the restriction that each spatial orbital should have two electrons, one with a and one with /3 spin, is normally made. Such wave functions are known as Restricted Hartree-Fock (RHF). Open-shell systems may also be described by restricted type wave functions, where the spatial part of the doubly occupied orbitals is forced to be the same this is known as Restricted Open-shell Hartree-Fock (ROHF). For open-shell species a UHF treatment leads to well-defined orbital energies, which may be interpreted as ionization potentials. Section 3.4. For an ROHF wave function it is not possible to chose a unitary transformation which makes the matrix of Lagrange multipliers in eq. (3.40) diagonal, and orbital energies from an ROHF wave function are consequently not uniquely defined, and cannot be equated to ionization potentials by a Koopman type argument. 

We obtain a physical interpretation of this approach by a suitable definition of the Lagrange multipliers /3U and /3jy- Thus we assume that... 

The constraints with the largest absolute Avalues are the ones whose right-hand sides affect the optimal value function V the most, at least for b close to b. However, one must account for the units for each bj in interpreting these values. For example, if some bj is measured in kilograms and both sides of the constraint hjix) = bj are multiplied by 2.2, then the new constraint has units of pounds, and its new Lagrange multiplier is 1/2.2 times the old one. 

In Eq. (2), jUtfd is the Lagrange multiplier taking care of the normalization constraint (3). It has the interpretation of the chemical potential of the electronic cloud and is written explicitly in Eq. (4) in the TFD approximation. It is seen from that equation that tfd can be viewed as built up from the sum of three different pieces ... 

This section presents first the formulation and basic definitions of constrained nonlinear optimization problems and introduces the Lagrange function and the Lagrange multipliers along with their interpretation. Subsequently, the Fritz John first-order necessary optimality conditions are discussed as well as the need for first-order constraint qualifications. Finally, the necessary, sufficient Karush-Kuhn-Dicker conditions are introduced along with the saddle point necessary and sufficient optimality conditions. 

The Lagrange multipliers in a constrained nonlinear optimization problem have a similar interpretation to the dual variables or shadow prices in linear programming. To provide such an interpretation, we will consider problem (3.3) with only equality constraints that is,... 

For the geometrical interpretation of the dual problem, we will consider particular values for the Lagrange multipliers i, fi2 associated with the two inequality constraints (fa > 0, fa > 0), denoted as fa, fa. 

Remark 3 If the primal problem (P) has an optimal solution and it is stable, then using the theorem of existence of optimal multipliers (see section 4.1.4), we have an alternative interpretation of the optimal solution (A, p) of the dual problem (D) that (A, p) are the optimal Lagrange multipliers of the primal problem (P). 

An alternative interpretation of this equation is that y(t) is a time-dependent Lagrange multiplier, introduced to enforce constant normalization By the... 

In the last equation X is the Lagrange multiplier and can be interpreted, analogously to the Slater transition state formula [26], as the effective electronegativity, or the negative of the chemical potential. Using the set of equations 31, the response of the charge deviation with respect to the external potential (u) measured relative to... 

The Lagrange multiplier has also a simple and instructive interpretation In the above examples it is a certain measure of the cost involved in the use of the operating policy. Thus it is easy to see in the first example that the greater the value of X the smaller will be the total sum Qfi(X) (this is shown in Fig. 2.3), for pi — ps+i — XX f is the profit from the transformation minus the cost of the operating variables In some cases we may have both a restriction... 

Such an interpretation of the Lagrange multiplier is especially useful for cases when both j. and have the same unites or they are unitless. For example, if f and a are variances of relative errors f/ and aja. In such a situation, a small value of the Lagrange multiplier logically is expected since a priori knowledge is always less certain than actual measurements. 

The truncation of the series expansion (4) at second order allows the solution to be found by solving a linear system of equations. The only complication is the need to enforce the constraint (2), which can be taken care of with the method of Lagrange multipliers. In this context, the Lagrange multiplier can be interpreted as the chemical potential, and the solution to the constrained problem is the charge distribution and chemical potential which minimizes the free energy... 

In the previous discussion, it will perhaps have become apparent that the generalized Lagrange multiplier or adjoint function plays a significant role in the theory of optimal processes. Furthermore, it becomes as necessary to solve the adjoint or costate equations as the state equations if we are to analyze or synthesize optimal systems. We have also noted that the adjoint functions appear in the Lagrangian as a weighting given to the source density 5. In this section, we shall take up this idea to develop a physical interpretation of the adjoint function which should help us understand its role and perhaps find the adjoint equations, boundary conditions, and even solutions more easily. This physical interpretation as an importance function follows closely the interpretation given to the adjoint function in reactor theory 54). 

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