ENTROPY OPTIMIZATION IN PHASE DETERMINATION WITH LINEAR INEQUALITY CONSTRAINTS

DETERMINATION WITH

LINEAR INEQUALITY CONSTRAINTS

VASILE PREDA, COSTEL B ˘ALC ˘AU and CRISTIAN NICULESCU

Using the Maximum Entropy Principle, we consider the phase problem in X-ray crystallography. The electron density distribution function is obtained from the knowledge of some lower and upper bounds for the components of the struc- ture factors. Then we analyze the problem as an entropy maximization problem with linear inequality constraints, which we solve by the geometric programming method. We obtain also a refined form of the solution when some average values of a prior distribution are given.

AMS 2000 Subject Classification: 94A17, 90C25, 90C46.

Key words: X-ray crystallography, maximum entropy, geometric programming method.

1. INTRODUCTION

The phase problem in X-ray crystallography is one of the most productive approaches to protein structure determination. When a protein is crystallized and an X-ray is applied to the crystal, then the crystal scatters this X-ray and produces X-ray diffractions (X-ray reflections). According to [8], different protein structures make different diffraction patterns and the crystallogra- phers use the record diffraction patterns to determine or distinguish between different protein structures.

A mathematical structure of protein crystal can be described by an elec- tron density distribution function ρ(r), where r is an arbitrary point in the crystal. Thus an X-ray diffraction can be represented by a complex number FH called the structure factor, where H is a three-dimensional integer vector serving as the index of the structure factor.

Following [2, 8, 15, 28], there is a direct mathematical relationship bet- ween the electron density distribution of the crystal and its structure factors:

the electron density distribution functionρcan be expanded as a Fourier series

REV. ROUMAINE MATH. PURES APPL.,55(2010),4, 327–340

with the structure factorsFH as the coefficients, namely, ρ(r) =X

H

FHexp(−2πiH^>r), FH= Z

V

ρ(r) exp(2πiH^>r)dr, where V ⊂R³ is the unit space of the crystal.

The structure factors are complex numbers, each having a magnitude and a phase. The magnitudes can be obtained by measuring the intensities of the diffraction spots, but the diffraction image does not contain any information about the phases. The phases need to be found for complete determination of the electron density distribution function of the crystal. Hence, we can enounce the following well-knownphase problem in X-ray crystallography, like as in [2, 8, 15, 28]: given the magnitudes of the structure factors, find the correct phases that define the electron density distribution function of the crystal system.

Hauptman and Karle [15] developed a method to solve directly the phase problem. This method is based on a nonlinear least-squares formulation of the problem and has been successfully applied only to small molecules, with less than 100 atoms. Bricogne and several others authors [2, 3, 4, 5, 6, 11] deve- loped a Bayesian statistical approach to the phase problem more suitable for large molecules such as proteins. This method consists in computation of the joint probability of the structure factors to determine the correct phases, us- ing the fact that the magnitudes of the structure factors are already known.

If

F_Hj, j ∈ {1,2, . . . , n} is a set of structure factors, considered as a set of random variables, then the joint probability of the structure factors is evalu- ated when they are assigned to some given values F_H^∗

j, j∈ {1,2, . . . , n}. This evaluation needs to be realized many times for many of the possible values as- signed to the structure factors. Based on the Maximum Entropy Principle from statistical mechanics and information theory (see, e.g., [13, 16, 17, 18, 19]), the joint probability of the structure factors is proportional to the maximum entropy of the crystal system with given structure factors. Wu, Phillips, Tapia and Zhang [28] have studied the entropy maximization problem in this ap- proach and developed a fast Newton algorithm for solving the phase problem.

In this paper, we extend the approach from [2, 28] to find the joint probability of the structure factors when we know only some lower and upper bounds for their components. For this, in Section 2, we define the problem of determination of the joint probability of the structure factors as an entropy optimization primal problem with linear inequality constraints. For solving this problem, in Section 3, we apply the geometric programming method (see, e.g., [9, 10, 21]) to derive a convex dual problem. Thus, we obtain some duality results relative to the formulated problems. Note that the entropy optimization and the geometric programming method have been used also to find probability

distributions under linear equality constraints [1, 12, 14, 22, 23]. In Section 4, we derive a refined form of the dual problem when some average values of the prior distribution are computed. Finally, in Section 5, we present three particular cases, when the dual problem can be further simplified.

2. PROBLEM STATEMENT

Throughout the paper we will denote by the componentwise order relation in the complex numbers set, i.e., if z1, z2∈Cwe write

z₁ z₂ iff Re(z₁)≤Re(z₂) and Im(z₁)≤Im(z₂).

Following the phase problem in X-ray crystallography, let V be the unit space of the crystal and let ρ be a normalized electron density distribution function on V. Let

FHj, j ∈ {1,2, . . . , n} be a set of structure factors and, for every j ∈ {1,2, . . . , n}, letfj1andfj2be some lower and upper given bounds, respec- tively, of the structure factor F_Hj. According to Maximum Entropy Principle (see, e.g., [13, 16, 17, 18, 19]), the joint probability of the structure factors can be obtained by solving the following entropy optimization problem

(P) :

maxS_m¯(ρ) =− Z

V

ρ(r) ln ρ(r)

¯

m(r)dr s.t.

f_j1 Z

V

ρ(r)C_j(r)drf_j2, ∀j∈ {1,2, . . . , n};

Z

V

ρ(r)dr = 1,

where ¯m is the uniform distribution on the unit spaceV of the crystal,S_m¯(ρ) is an entropy function,

C_j(r) = exp (2πiH^>_jr), ∀j∈ {1,2, . . . , n},

and, for any j ∈ {1,2, . . . , n}, the bounds fj1 and fj2 are known complex numbers.

We assume that the problem (P) is consistent.

We note that the distribution ¯mcan be interpreted as a prior given dis- tribution function containing a prior information about the crystal structure.

Also we remark that the function

−S_m¯(ρ) = Z

V

ρ(r) ln ρ(r)

¯ m(r)dr

is called the relative entropy of ρ with respect to m¯ (the cross-entropy of m¯ with respect to ρ; the Kullback-Leibler number). The objective functionS_m¯ of problem (P) is a strictly concave function (see, e.g., [16]) and the constraints are linear. It follows that the problem (P) has an unique optimal solution.

3. DUALITY

Throughout the rest of this paper we will denote by·the componentwise product (inner product) of complex numbers, i.e., if z1, z2∈Cwe write

z₁·z₂ = Re(z₁)Re(z₂) + Im(z₁)Im(z₂).

Using the geometric programming method (see, e.g., [9, 10, 21]), we can define a convex dual problem for problem (P) as

(D) :

minD(λ) = lnZ(λ)−

n

X

j=1

λj1·fj1+

n

X

j=1

λj2·fj2 s.t.

Z

V

m(r)U¯ (λ,r)[Cj(r)−fj1]dr0, ∀j∈ {1,2, . . . , n};

Z

V

m(r)U¯ (λ,r)[C_j(r)−f_j2]dr0, ∀j∈ {1,2, . . . , n};

λj1 0, λj2 0, ∀j∈ {1,2, . . . , n},

where for all j ∈ {1,2, . . . , n}, λ_j1 and λ_j2 are complex numbers (Lagrange multipliers), λ= (λ11, λ12, λ21, λ22, . . . , λn1, λn2),

U(λ,r) = exp





n

X

j=1

(λj1−λj2)·Cj(r)



, and

Z(λ) = Z

V

m(r)U¯ (λ,r)dr.

Since the Hessian of the dual objective function D is positive definite (see [28]), then Dis a strictly convex function. Hence, if the dual problem (D) has an interior feasible solution, according to the Fenchel duality (see, e.g., [26]) it follows that the dual problem (D) has an unique optimal solution.

Now we provide the following duality theorems.

Theorem 3.1 (Weak duality). If ρ and λ are feasible solutions of prob- lems (P) and (D), respectively, then

(1) S_m¯(ρ)≤ D(λ).

Moreover, the equality holds only if

(2) ρ(r) = m(r)U¯ (λ,r)

Z(λ) , ∀r∈V.

Proof. By the definitions of problems (P) and (D) and using the Jensen’s inequality for the concave function lnxwe have

S_m¯(ρ) +

n

X

j=1

λ_j1·f_j1−

n

X

j=1

λ_j2·f_j2≤ Z

V

ρ(r) lnm(r)¯ ρ(r)dr+

+

n

X

j=1

(λj1−λj2)· Z

V

ρ(r)Cj(r)dr = Z

V

ρ(r) lnm(r)U¯ (λ,r) ρ(r) dr≤

≤ln Z

V

ρ(r)m(r)U¯ (λ,r)

ρ(r) dr = lnZ(λ).

Inequality (1) is proved. Applying the equality part of Jensen’s inequality we obtain that inequality (1) becomes an equality only if

m(r)U(λ,¯ r)

ρ(r) =K, ∀r∈V, K being a constant with respect to r. Since R

V ρ(r)dr = 1, we getK =Z(λ), hence equality (2) is also proved.

Theorem 3.2 (Strong duality). If λ^∗ = (λ^∗₁₁, λ^∗₁₂, λ^∗₂₁, λ^∗₂₂, . . . , λ^∗_n1, λ^∗_n2) is an interior optimal solution of problem (D), then ρ^∗ defined by

(3) ρ^∗(r) = m(r)U¯ (λ^∗,r)

Z(λ^∗) , ∀r∈V

is the optimal solution of problem (P) and the duality gap vanishes, i.e., S_m¯(ρ^∗) =D(λ^∗).

Proof. Obviously, (4)

Z

V

ρ^∗(r)dr = 1.

For all j∈ {1,2, . . . , n} we have Z

V

ρ^∗(r)C_j(r)dr = Z

V

m(r)U¯ (λ^∗,r)Cj(r)dr Z(λ^∗)

Z

V

m(r)U(λ¯ ^∗,r)fj1dr Z(λ^∗) =f_j1. In the same way, it follows that

Z

V

ρ^∗(r)C_j(r)drf_j2, ∀j∈ {1,2, . . . , n}, hence ρ^∗ is a feasible solution of problem (P).

We shall show, by contradiction, that for allj ∈ {1,2, . . . , n}the equality

(5) λ^∗_j1·

Z

V

m(r)U¯ (λ^∗,r)[Cj(r)−fj1]dr = 0

holds. Suppose that

(6) λ^∗_j1·

Z

V

m(r)U¯ (λ^∗,r)[C_j(r)−f_j1]dr>0.

Since λ^∗ is an interior dual feasible solution we have _∂λ^∂D

j1(λ^∗) = 0, hence Z

V

¯

m(r)U(λ^∗,r)[C_j(r)−f_j1]dr = 0.

But this contradicts (6). Thus (5) holds. In the same manner we can prove that for all j∈ {1,2, . . . , n} we also have

(7) λ^∗_j2·

Z

V

m(r)U¯ (λ^∗,r)[Cj(r)−fj2]dr = 0.

Using (3), (4), (5) and (7) we obtain that

S_m¯(ρ^∗) =− Z

V

ρ^∗(r) ln exp

"

n

P

j=1

(λ^∗_j1−λ^∗_j2)·Cj(r)

#

Z(λ^∗) dr = (8)

=−

n

X

j=1

(λ^∗_j1−λ^∗_j2)· Z

V

ρ^∗(r)C_j(r)dr + lnZ(λ^∗) Z

V

ρ^∗(r)dr =D(λ^∗).

On the other hand, ifρ is an arbitrary feasible solution of problem (P), in the same manner as above we obtain that

Z

V

ρ(r) lnρ^∗(r)

¯

m(r)dr =

n

X

j=1

(λ^∗_j1−λ^∗_j2)· Z

V

ρ(r)Cj(r)dr−lnZ(λ^∗)≥ (9)

≥

n

X

j=1

λ^∗_j1·f_j1−

n

X

j=1

λ^∗_j2·f_j2−lnZ(λ^∗) =−D(λ^∗).

It follows from (8) and (9) that Z

V

ρ^∗(r) lnρ^∗(r)

¯

m(r)dr≤ Z

V

ρ(r) lnρ^∗(r)

¯ m(r)dr,

for every feasible solution ρ of problem (P). Therefore, by Csisz´ar characteri- zation of minimum relative entropy [7], we conclude that ρ^∗ is an optimal solution of problem (P).

Remark 3.1. By taking

f_j1 =f_j2=f_j, ∀j∈ {1,2, . . . , n}, and noting

λj =λj1−λj2, ∀j ∈ {1,2, . . . , n},

we retrive the duality results from [28] concerning the entropy maximization in phase determination with linear equality constraints. In this particular case, we can remove all the constraints of the dual problem (D), these constraints being satisfied by the dual optimal solution since

∂D

∂λ_j(λ^∗) = 0, ∀j ∈ {1,2, . . . , n}.

Therefore, in this case the dual problem (D) will be an unconstrained convex minimization problem.

4. A STRAIGHTFORWARD FORM OF THE DUAL If the average valuesR

V m(r)C¯ j(r)dr,j∈ {1,2, . . . , n}of the prior proba- bility distribution ¯m are known, then we can obtain a simplified form of the dual problem (D).

Denote J0·=

j∈ {1,2, . . . , n} |Re(f_j1)≤Re Z

V

¯

m(r)C_j(r)dr

≤Re(f_j2)

,

J1·=

j∈ {1,2, . . . , n} |Re Z

V

¯

m(r)Cj(r)dr

<Re(fj1)

,

J2·=

j∈ {1,2, . . . , n} |Re Z

V

¯

m(r)Cj(r)dr

>Re(fj2)

,

J·0=

j∈ {1,2, . . . , n} |Im(f_j1)≤Im Z

V

¯

m(r)C_j(r)dr

≤Im(f_j2)

,

J·1=

j∈ {1,2, . . . , n} |Im Z

V

¯

m(r)Cj(r)dr

<Im(fj1)

,

J·2=

j∈ {1,2, . . . , n} |Im Z

V

¯

m(r)C_j(r)dr

>Im(f_j2)

, J_st=Js·∩J·t, ∀s, t∈ {0,1,2}.

We note that the set{1,2, . . . , n} have the following three partitions

{1,2, . . . , n}=

2

[

s=0

Js·=

2

[

t=0

J·t=

2

[

s=0 2

[

t=0

J_st.

The dual problem (D) has now the simplified form

(D1) :

minD₁(α, β, γ, δ) = lnZ₁(α, β, γ, δ)− X

j∈J1·

α_jRe(f_j1)−

− X

j∈J·1

β_jIm(f_j1) + X

j∈J2·

γ_jRe(f_j2) + X

j∈J·2

δ_jIm(f_j2) s.t.

Z

V

m(r)U¯ ₁(α, β, γ, δ,r)Re[C_j(r)−f_j1]dr≥0, ∀j∈J0·, Z

V

m(r)U¯ ₁(α, β, γ, δ,r)Im[Cj(r)−fj1]dr≥0, ∀j∈J·0, Z

V

¯

m(r)U₁(α, β, γ, δ,r)Re[C_j(r)−f_j2]dr≤0, ∀j∈J0·, Z

V

m(r)U¯ ₁(α, β, γ, δ,r)Im[C_j(r)−f_j2]dr≤0, ∀j∈J·0, α, β, γ, δ≥0,

whereαj,βj,γj andδj are real numbers (Lagrange multipliers),α= (αj)j∈J1·, β = (β_j)j∈J·1,γ= (γ_j)j∈J2·,δ = (δ_j)j∈J·2,

U₁(α, β, γ, δ,r) = exp

"

X

j∈J1·

αjRe[Cj(r)] + X

j∈J·1

βjIm[Cj(r)]−

− X

j∈J2·

γjRe[Cj(r)]− X

j∈J·2

δjIm[Cj(r)]

# , and

Z₁(α, β, γ, δ) = Z

V

m(r)U¯ ₁(α, β, γ, δ,r)dr.

Remark 4.1. Problem (D1) has only 2n−|J0·|−|J·0|real variables. Thus, by changing (D) into (D1) we reduce both the number of variables and the number of constraints of dual problem.

Next, we state and prove the duality theorems.

Theorem4.1 (Weak duality). If ρ and (α, β, γ, δ) are feasible solutions of problem (P) and (D₁), respectively, then

(10) S_m¯(ρ)≤ D₁(α, β, γ, δ).

Moreover, the equality holds only if

(11) ρ(r) = m(r)U¯ ₁(α, β, γ, δ,r)

Z₁(α, β, γ, δ) , ∀r∈V.

Proof.By the definitions of problems (P) and (D1) and using the Jensen’s inequality for the concave function lnxwe have

S_m¯(ρ) + X

j∈J1·

α_jRe(f_j1) + X

j∈J·1

β_jIm(f_j1)− X

j∈J2·

γ_jRe(f_j2)− X

j∈J·2

δ_jIm(f_j2)≤

≤ Z

V

ρ(r) lnm(r)¯

ρ(r)dr + X

j∈J1·

α_jRe Z

V

ρ(r)C_j(r)dr

+

+ X

j∈J·1

β_jIm Z

V

ρ(r)C_j(r)dr

− X

j∈J2·

γ_jRe Z

V

ρ(r)C_j(r)dr

−

− X

j∈J·2

δ_jIm Z

V

ρ(r)C_j(r)dr

= Z

V

ρ(r) lnm(r)U¯ ₁(α, β, γ, δ,r)

ρ(r) dr≤

≤ln Z

V

ρ(r)m(r)U¯ ₁(α, β, γ, δ,r)

ρ(r) dr = lnZ₁(α, β, γ, δ).

Inequality (10) is proved. Applying the equality part of Jensen’s inequality we obtain that inequality (10) becomes an equality only if the equality (11) holds.

Theorem4.2 (Strong duality). Let(α^∗, β^∗, γ^∗, δ^∗)be an interior optimal solution of problem (D₁). Then ρ^∗ defined by

(12) ρ^∗(r) = m(r)U¯ ₁(α^∗, β^∗, γ^∗, δ^∗,r)

Z₁(α^∗, β^∗, γ^∗, δ^∗) , ∀r∈V

is the optimal solution of problem (P) and the duality gap vanishes, i.e., (13) S_m¯(ρ^∗) =D₁(α^∗, β^∗, γ^∗, δ^∗).

Proof. Obviously, (14)

Z

V

ρ^∗(r)dr = 1.

For every j ∈ J1·, (α^∗, β^∗, γ^∗, δ^∗) is an interior dual feasible solution with respect to the component α_j, hence ^∂D_∂α¹

j(α^∗, β^∗, γ^∗, δ^∗) = 0. It follows that Z

V

¯

m(r)U₁(α^∗, β^∗, γ^∗, δ^∗,r)Re[C_j(r)]dr

Z₁(α^∗, β^∗, γ^∗, δ^∗) −Re(fj1) = 0, hence

(15) Re

Z

V

ρ^∗(r)C_j(r)dr

= Re(f_j1), ∀j∈J1·.

In the same way, it follows that

(16) Im

Z

V

ρ^∗(r)Cj(r)dr

= Im(fj1), ∀j ∈J·1,

(17) Re

Z

V

ρ^∗(r)C_j(r)dr

= Re(f_j2), ∀j∈J2·,

(18) Im

Z

V

ρ^∗(r)Cj(r)dr

= Im(fj2), ∀j ∈J·2.

For every j∈J0·, using the definition of problem (D1) we obtain that

Re Z

V

ρ^∗(r)Cj(r)dr

= Z

V

¯

m(r)U₁(α^∗, β^∗, γ^∗, δ^∗,r)Re[C_j(r)]dr Z₁(α^∗, β^∗, γ^∗, δ^∗) ≥

≥ Z

V

m(r)U¯ ₁(α^∗, β^∗, γ^∗, δ^∗,r)Re(fj1)dr Z₁(α^∗, β^∗, γ^∗, δ^∗) , hence

(19) Re

Z

V

ρ^∗(r)C_j(r)dr

≥Re(f_j1), ∀j ∈J0·. In the same way, it follows that

(20) Im

Z

V

ρ^∗(r)Cj(r)dr

≥Im(fj1), ∀j∈J·0,

(21) Re

Z

V

ρ^∗(r)Cj(r)dr

≤Re(fj2), ∀j ∈J0·,

(22) Im

Z

V

ρ^∗(r)C_j(r)dr

≤Im(f_j2), ∀j∈J·0. It follows from (15), (17), (19) and (21) that

(23) Re(fj1)≤Re Z

V

ρ^∗(r)Cj(r)dr

≤Re(fj2), ∀j∈ {1,2, . . . , n}.

It follows from (16), (18), (20) and (22) that (24) Im(fj1)≤Im

Z

V

ρ^∗(r)Cj(r)dr

≤Im(fj2), ∀j∈ {1,2, . . . , n}.

From (23) and (24) we deduce that fj1

Z

V

ρ^∗(r)Cj(r)drfj2, ∀j∈ {1,2, . . . , n},

hence ρ^∗ is a feasible solution of problem (P).

Using (12), (14), (15), (16), (17) and (18) we obtain that S_m¯(ρ^∗) =−

Z

V

ρ^∗(r) lnU₁(α^∗, β^∗, γ^∗, δ^∗,r) Z₁(α^∗, β^∗, γ^∗, δ^∗) dr =

=− X

j∈J1·

Z

V

ρ^∗(r)α^∗_jRe[C_j(r)]dr− X

j∈J·1

Z

V

ρ^∗(r)β_j^∗Im[C_j(r)]dr+

+ X

j∈J2·

Z

V

ρ^∗(r)γ_j^∗Re[Cj(r)]dr + X

j∈J·2

Z

V

ρ^∗(r)δ_j^∗Im[Cj(r)]dr+

+ lnZ₁(α^∗, β^∗, γ^∗, δ^∗) Z

V

ρ^∗(r)dr =

=− X

j∈J1·

α^∗_jRe Z

V

ρ^∗(r)C_j(r)

dr− X

j∈J·1

β_j^∗Im Z

V

ρ^∗(r)C_j(r)

dr+

+ X

j∈J2·

γ_j^∗Re Z

V

ρ^∗(r)C_j(r)

dr + X

j∈J·2

δ^∗_jIm Z

V

ρ^∗(r)C_j(r)

dr+

+ lnZ₁(α^∗, β^∗, γ^∗, δ^∗) =D₁(α^∗, β^∗, γ^∗, δ^∗).

On the other hand, ifρ is an arbitrary feasible solution of problem (P), in the same manner as above we obtain

Z

V

ρ(r) lnρ^∗(r)

¯

m(r)dr = Z

V

ρ(r) lnU₁(α^∗, β^∗, γ^∗, δ^∗,r) Z₁(α^∗, β^∗, γ^∗, δ^∗) dr =

= X

j∈J1·

Z

V

ρ(r)α^∗_jRe[C_j(r)]dr + X

j∈J·1

Z

V

ρ(r)β_j^∗Im[C_j(r)]dr−

− X

j∈J2·

Z

V

ρ(r)γ_j^∗Re[Cj(r)]dr− X

j∈J·2

Z

V

ρ(r)δ_j^∗Im[Cj(r)]dr−

−lnZ₁(α^∗, β^∗, γ^∗, δ^∗) Z

V

ρ(r)dr =

= X

j∈J1·

α^∗_jRe Z

V

ρ(r)C_j(r)

dr + X

j∈J·1

β_j^∗Im Z

V

ρ(r)C_j(r)

dr−

− X

j∈J2·

γ_j^∗Re Z

V

ρ(r)C_j(r)

dr− X

j∈J·2

δ^∗_jIm Z

V

ρ(r)C_j(r)

dr−

−lnZ₁(α^∗, β^∗, γ^∗, δ^∗)≥ −D₁(α^∗, β^∗, γ^∗, δ^∗).

It follows that Z

V

ρ^∗(r) lnρ^∗(r)

¯

m(r)dr≤ Z

V

ρ(r) lnρ^∗(r)

¯ m(r)dr,

for every feasible solution ρ of problem (P). Therefore, by Csisz´ar characte- rization of minimum relative entropy [7], we conclude that ρ^∗ is an optimal solution of problem (P).

5. SOME PARTICULAR CASES Case 1. If

J0·=J·0 ={1,2, . . . , n},

then the prior probability distribution ¯mis a feasible solution for primal prob- lem (P). Using the following well-known property of relative entropy (see, e.g., [16])

S_m¯(ρ) =− Z

V

ρ(r) ln ρ(r)

¯

m(r)dr≤0 =S_m¯( ¯m)

we deduce that the prior probability distribution ¯mis the optimal solution of problem (P), namely, ρ^∗ = ¯m.

Case 2. If

J0·=J·0=∅, then the dual problem (D1) has the form

minD₁(α, β, γ, δ) s.t.

α, β, γ, δ ≥0.

Hence, in this case the dual problem (D1) has only the nonnegativity con- straints.

Case 3. If

J1·={k}, J0·={1,2, . . . , n} \ {k}, J·0={1,2, . . . , n}, where k∈ {1,2, . . . , n}, then the dual problem (D₁) has the form

minD₁(α_k) = ln Z

V

m(r) exp{α¯ _kRe[C_k(r)]}dr−α_kRe(f_k1) s.t.

Z

V

m(r) exp{α¯ _kRe[C_k(r)]}Re[C_j(r)−f_j1]dr≥0, ∀j ∈ {1,2, . . . , n} \ {k}, Z

V

m(r) exp{α¯ _kRe[C_k(r)]}Im[C_j(r)−fj1]dr≥0, ∀j∈ {1,2, . . . , n}, Z

V

m(r) exp{α¯ _kRe[C_k(r)]}Re[C_j(r)−f_j2]dr≤0, ∀j ∈ {1,2, . . . , n} \ {k}, Z

V

m(r) exp{α¯ _kRe[C_k(r)]}Im[C_j(r)−fj2]dr≤0, ∀j∈ {1,2, . . . , n},

where αk ∈ R. Hence, in this case the dual problem (D1) has a single real variable. Also, we can remove the nonnegativity restrictionα_k≥0 of the dual problem, this constraint being satisfied by the dual optimal solution.

Remark 5.1. Similarly to Case 3, we can derive a simplified form of the dual problem (D₁) in the cases

Case 4. J2·={k}, J0·={1,2, . . . , n} \ {k}, J·0={1,2, . . . , n};

Case 5. J0·={1,2, . . . , n}, J_·1 ={k}, J_·0={1,2, . . . , n} \ {k};

Case 6. J0·={1,2, . . . , n}, J_·2 ={k}, J_·0={1,2, . . . , n} \ {k}, where k∈ {1,2, . . . , n}.

REFERENCES

[1] C. B˘alc˘au,Construction of some probability distributions by maximum entropy principle.

An. Univ. Bucure¸sti Mat. Inform.51(2002), 99–108.

[2] G. Bricogne,Maximum entropy and the foundations of direct methods. Acta Cryst. Sect.

A40(1984), 410–445.

[3] G. Bricogne, A Bayesian statistical theory of the phase problem. I. A multichannel maximum-entropy formalism for constructing generalized joint probability distributions of structure factors. Acta Cryst. Sect. A44(1988), 517–545.

[4] G. Bricogne,A multisolution method of phase determination by combined maximization of entropy and likelihood.III.Extension to powder diffraction data. Acta Cryst. Sect. A 47(1991), 803–829.

[5] G. Bricogne,Bayesian statistical viewpoint on structure determination: Basic concepts and examples. Methods in Enzymology 276(1997), 361–423.

[6] G. Bricogne and C.J. Gilmore,A multisolution method of phase determination by com- bined maximization of entropy and likelihood.I.Theory, algorithms and strategy. Acta Cryst. Sect. A46(1990), 284–297.

[7] I. Csisz´ar,I-divergence geometry of probability distributions and minimization problems.

Ann. Probability3(1975), 146–158.

[8] J. Drenth, Principles of Protein X-Ray Crystallography. Springer-Verlag, New York, 1994.

[9] S. Erlander,Entropy in linear programs. Math. Programming21(1981), 137–151.

[10] S.-C. Fang, J.R. Rajasekera and H.-S.J. Tsao,Entropy Optimization and Mathematical Programming. Kluwer, Boston, 1997.

[11] E.L. Fortelle and G. Bricogne,Maximum-likelihood heavy-atom parameter refinement for multiple isomorphous replacement and multiwavelength anomalous diffraction methods.

Methods in Enzymology276(1997), 472–494.

[12] S. Guia¸su, A classification of the main probability distributions by minimizing the weighted logarithmic measure of deviation. Ann. Inst. Statist. Math. 42 (1990), 269–

279.

[13] S. Guia¸su,Quantum Mechanics. Nova Science Publ, Huntington, New York, 2001.

[14] H. Gzyl,Maxentropic reconstruction of some probability distributions. Stud. Appl. Math.

105(2000), 235–243.

[15] H. Hauptman and J. Karle, Solution of the Phase Problem. I. The Centrosymmetric Crystal. ACA Monograph3, American Crystallographic Association, New York, 1953.

[16] S. Ihara,Information Theory for Continuous Systems. World Scientific Publ., Singapore, 1993.

[17] E.T. Jaynes, Information theory and statistical mechanics. Phys. Rev.106,2 (1957), 620–630.

[18] E.T. Jaynes,Information theory and statistical mechanics.II. Phys. Rev.108,2(1957), 171–190.

[19] S. Kullback and R.A. Leibler, On information and sufficiency. Ann. Math. Statistics 22(1951), 79–86.

[20] C. Niculescu,Optimality and duality in multiobjective fractional programming involving ρ-semilocally type I-preinvex and related functions. J. Math. Anal. Appl. 335 (2007), 7–19.

[21] E.L. Peterson,Geometric programming. SIAM Rev.18(1976), 1–51.

[22] V. Preda, The Student distribution and the principle of maximum entropy. Ann. Inst.

Statist. Math.34(1982), 335–338.

[23] V. Preda and C. B˘alc˘au, On maxentropic reconstruction of countable Markov chains and matrix scaling problems. Stud. Appl. Math.111(2003), 85–100.

[24] V. Preda and C. B˘alc˘au, Maxentropic reconstruction of some probability distributions with linear inequality constraints. In:Proceedings of the7th Balkan Conference on Opera- tional Research(Constant¸a, 2005), pp. 151–158, Bucharest, 2007.

[25] V. Preda, I.M. Stancu-Minasian, M. Beldiman and A.M. Stancu, Generalized V- univexity type-I for multiobjective programming withn-set functions. J. Global Optim.

44(2009), 131–148.

[26] R.T. Rockafellar,Convex Analysis. Princeton Univ. Press., Princeton, 1970.

[27] I.M. Stancu-Minasian, Optimality and duality in nonlinear programming involving semilocally B-preinvex and related functions. European J. Oper. Res.173(2006), 47–58.

[28] Z. Wu, G. Phillips, R. Tapia and Y. Zhang,A fast Newton algorithm for entropy maxi- mization in phase determination. SIAM Rev.43(2001), 623–642.

Received 5 May 2010 University of Bucharest

Faculty of Mathematics and Computer Science Str. Academiei 14, 010014 Bucharest, Romania

preda@fmi.unibuc.ro

&“Gheorghe Mihoc–Caius Iacob” Institute of Mathematical Statistics and Applied Mathematics

Calea 13 Septembrie nr. 13 050711 Bucharest 5, Romania

University of Pite¸sti

Faculty of Mathematics and Computer Science Str. Tˆargu din Vale 1, 110440 Pite¸sti, Romania

costel.balcau@upit.ro and

University of Bucharest

Faculty of Mathematics and Computer Science Str. Academiei 14, 010014 Bucharest, Romania

crnicul@fmi.unibuc.ro