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Global minimization using an Augmented Lagrangian method with variable lower-level constrai

Global minimization using an Augmented Lagrangian method

with variable lower-level constraints

E.G.Birgin? C.A.Floudas?J.M.Mart′?nez?

January22,2007

Abstract

A novel global optimization method based on an Augmented Lagrangian framework is

introduced for continuous constrained nonlinear optimization problems.At each outer iter-

ation k the method requires theεk-global minimization of the Augmented Lagrangian with

simple constraints,whereεk→ε.Global convergence to anε-global minimizer of the orig-

inal problem is proved.The subproblems are solved using theαBB method.Numerical

experiments are presented.

Key words:deterministic global optimization,Augmented Lagrangians,nonlinear pro-

gramming,algorithms,numerical experiments.

1Introduction

Global optimization has ubiquitous applications in all branches of engineering sciences and applied sciences.During the last decade several textbooks addressed di?erent facets of global optimization theory and applications[10,15,18,25,44,53,54,57].Recent review papers have also appeared[16,34].

The Augmented Lagrangian methodology based on the Powell-Hestenes-Rockafellar[24,37, 39]formula has been successfully used for de?ning practical nonlinear programming algorithms [6,7,12,14].Convergence to KKT points was proved using the Constant Positive Linear Dependence constraint quali?cation[5],which strengthens the results based on the classical regularity condition[11,14].

In this work,we consider the Augmented Lagrangian method introduced in [7]and we modify it in such a way that,at each outer iteration,we ?nd an ε-global minimizer of the subproblem.In the de?nition of the subproblem we introduce an important modi?cation with respect to [7]:besides the lower level constraints we include constraints that incorporate information about the global solution of the nonlinear programming problem.A theorem of convergence to ε-global minimizers is presented.

In the implementation,we consider linear constraints on the lower-level set,and additional valid linear constraints which result from outer approximations of the feasible region and hence incorporate the global optimum information.This allows us to use the αBB [1,2,3,9]method and its convex underestimation techniques [31,32]for the subproblems,in such a way that the underestimation techniques are applied just to the Augmented Lagrangian function and not to the constraints.The αBB global optimization approach has been applied to various problems that include molecular conformations in protein folding,parameter estimation and phase equilibrium.Mixed-integer nonlinear models arising in process synthesis,design and operations problems represent additional important application areas.See [15]and the references therein for details.

It is important to emphasize that there exist many global optimization techniques for non-linear programming problems,e.g.,[2,3,4,9,17,19,20,21,22,23,27,28,36,41,42,45,46,47,48,49,50,55].However,to our knowledge,none of them is based on Augmented Lagrangians.Moreover,as a consequence of using the Augmented Lagrangian approach combined with the αBB method and its convex α-underestimation techniques,the method introduced in this paper does not rely on the speci?c form of the functions involved in the problem de?nition (objective function and constraints),apart from their continuity and di?erentiability.Interval arithmetic is used to compute bounds on the objective function and to compute the convex α-underestimators.Although the method can take advantage of known underestimators and relaxations for several kinds of functional forms,it can also deal with functional forms for which underestimators and relaxations have not been developed yet.In this sense,the method does not depend on the analysis of expressions involved in the problem de?nition to identify functional forms for which ad-hoc underestimators are available.

This paper is organized as follows.In Section 2we describe the Augmented Lagrangian deterministic global optimization algorithm.The convergence to ε-global minimizers is proved in Section 3.In Section 4we describe the global optimization of the Augmented Lagrangian subproblems.Numerical results are given in Section 5.In Section 6we draw some conclusions.Notation.

If v ∈I R n ,v =(v 1,...,v n ),we denote v +=(max {0,v 1},...,max {0,v n }).If K =(k 1,k 2,...)?I N (with k j

I N .

The symbol · will denote the Euclidian norm.

2The overall algorithm

The problem to be addressed is:

Minimize f(x)

subject to h(x)=0

g(x)≤0

x∈?

(1)

where h:I R n→I R m,g:I R n→I R p,f:I R n→I R are continuous and??I R n is closed.A typical set?consists of“easy”constraints such as linear constraints and box constraints.By easy we mean that a suitable algorithm for local minimization is available.

Assumption1.From now on we will assume that there exists a global minimizer z of the problem.

We de?ne the following Augmented Lagrangian function:

Lρ(x,λ,μ)=f(x)+ρ

i=1 max 0,g i(x)+μi

ρk ,i=1,...,p.

If k=1or

max{ h(x k) ∞, V k ∞}≤τmax{ h(x k?1) ∞, V k?1 ∞},(4) de?neρk+1=ρk.Otherwise,de?neρk+1=γρk.

Step https://www.wendangku.net/doc/0b17638828.html,pute λk +1

∈[λmin ,λmax ],i =1,...,m and μk +1i ∈[0,μmax ],i =1,...,p .Set k ←k +1and go to Step 1.Remark.In the implementation,we will compute λk +1i

=min {max {λmin ,λk i +ρh i (x k )},λmax }and μk +1

=min {max {0,μk +ρg i (x k )},μmax }.These de?nitions correspond to safeguarded choices of ?rst-order Lagrange multiplier estimates.After the resolution of each subproblem,the vectors λk /ρk and μk /ρk represent shifts of the origin with respect to which infeasibility is penalized.The intutitive idea is that these shifts “correct”the previous decision on the best possible origin and enhance the possibility of achieving feasibility at the present iteration.The theoretical consequence is that,under suitable assumptions,one is able to prove that the penalty parameter does not need to go to in?nity [7].In practice,this implies that the subproblems tend to remain well conditioned.

We emphasize that the deterministic global optimization method αBB will not use the point x k ?1

as “initial approximation”as most local optimization solvers do.In fact,the concept of “initial point”has no meaning at all in this case.The information used by the outer iteration k is the set of approximate Lagrange multipliers computed after iteration k ?1,and nothing else.3Convergence to an ε-global minimum

In the theorems that follow,we assume that the sequence {x k }is well de?ned.In other words,the εk -global minimizer of the Augmented Lagrangian can always be found.A su?cient con-dition on the problem that guarantees that this assumption holds is the compactness of ?.In practical Optimization it is usual to add box constraints to the feasible set of the problem that re?ect some previous knowledge on the localization of the solution.Clearly,after intersection with a box,the feasible set becomes compact.

Theorem 1.Assume that the sequence {x k }is well de?ned and admits a limit point x ?.Then,x ?is feasible.

Proof.Since ?is closed and x k ∈?,we have that x ?∈?.We consider two cases:{ρk }bounded and ρk →∞.If {ρk }is bounded,there exists k 0such that ρk =ρk 0for all k ≥k 0.Therefore,for all k ≥k 0,(4)holds.This implies that h (x k ) →0and V k →0.So,g i (x k )+→0for all i =1,...,p .So,the limit point is feasible.

Now,assume that ρk →∞.Let z be as in Step 1.Therefore,z is feasible.So, h (z ) = g (z )+ =0.Suppose,by contradiction,that x ?is not feasible.Therefore,

h (x ?) 2+ g (x ?)+ 2> h (z ) 2+ g (z )+ 2.

Let K be an in?nite sequence of indices such that lim k ∈K x k =x ?.Since h and g are continuous,λk ,μk are bounded and ρk →∞,there exists c >0such that for k ∈K large enough:

h (x k )+

λk ρk + 2> h (z )+λk ρk +

2+c.4

Therefore,

f(x k)+ρk

ρk

g(x k)+μk

h(z)+λkρk

+ρk c

+f(x k)?f(z)>εk.

Therefore,

f(x k)+ρk

ρk

g(x k)+μk2

h(z)+λkρk

+εk.

Now,since z is a global minimizer,we have that z∈?∩P k for all k.Therefore,the inequality above contradicts the de?nition of x k. Theorem2.Under the same assumptions of Theorem1,every limit point x?of a sequence {x k}generated by Algorithm1is anε-global minimizer of the problem.

Proof.Let K?

∞I N be such that lim k∈K x k=x?.By Theorem1,x?is feasible.Let z∈?as as

in Step1.Then,z∈P k for all k.

We consider two cases:ρk→∞and{ρk}bounded. Case1(ρk→∞):By the de?nition of the algorithm:

f(x k)+ρk

ρk

g(x k)+μk2

h(z)+λkρk

+εk

(5)

for all k∈I N.

Since h(z)=0and g(z)≤0,we have:

h(z)+λkρk

2and

g(z)+μkρ

Therefore,by(5),

f(x k)≤f(x k)+ρk

ρk

g(x k)+μk2ρ

+ μk 2

Since z is a global minimizer,it turns out that x ?is an ε-global minimizer,as we wanted to prove.Case 2({ρk }bounded):In this case,we have that ρk =ρk 0for all k ≥k 0.Therefore,by the de?nition of Algorithm 1,we have:

f (x k )+ρk 0

ρk 0 2+ g (x k )+μk 2 h (z )+λk ρk 0 +

2 +εk for all k ≥k 0.Since g (z )≤0and μk /ρk 0≥0,

g (z )+μk ρk 0

.Thus,since h (z )=0,f (x k )+

ρk 0

2+ g (x k )+

μk 2

λk ρk 0

+εk for all k ≥k 0.Let K 1?∞

K be such that

lim k ∈K 1

λk =λ?,lim k ∈K 1

μk =μ?.

By the feasibility of x ?,taking limits in the inequality above for k ∈K 1,we get:

f (x ?)+ρk 0

ρk 0 2+ g (x ?)+μ?2 λ?ρk 0

2 +ε.Therefore,

f (x ?)+

ρk 0

ρk 0 + 2

≤f (z )+ρk 0ρk 0 2+ε.Thus,

f (x ?)+

ρk 0

≤f (z )+

ρk 0

+ε.(6)

Now,if g i (x ?)=0,since μ?i /ρk 0≥0,we have that

g i (x ?

)+μ?i ρk 0

Therefore,by (6),

f (x ?)+

ρk 0

≤f (z )+

ρk 0

+ε.(7)

But,by Step 2of Algorithm 1,lim k ∈∞max {g i (x k ),?μk i /ρk 0}=0.Therefore,if g i (x ?)<0we

necessarily have that μ?i =0.Therefore,(7)implies that f (x ?)≤f (z )+ε.Since z is a global

minimizer,the proof is complete.

4Global optimization of subproblems

In this section,we address the problem of?nding x k∈?∩P k satisfying(3)and we restrict ourselves to the case in which?is de?ned by linear constraints.This problem is equivalent to the problem of?nding anεk-global solution of the problem:

Minimize Lρ

(x,λk,μk)subject to x∈?∩P k,(8) where?={x∈I R n|Ax=b,Cx≤d,l≤x≤u}and Ax=b,Cx≤d and l≤x≤u represent the linear equality,linear inequality and bound constraints of problem(1),respectively.The remaining constraints of problem(1)will be h(x)=0,g(x)≤0.The role of P k will be elucidated soon.

To solve problem(8),we introduced and implemented theαBB algorithm[2,3]for the particular case of linear constraints and bounds.TheαBB method is a deterministic global optimization method for nonlinear programming problems based on Branch&Bound.For bounding purposes,it uses the convexα-underestimator of the function being minimized that coincides with the function at the bounds of the box and whose maximum separation(distance to the objective function)is proportional to the box dimensions.Therefore,the smaller the box, the tighter the convexα-underestimator.

Based on the last observation,theαBB method consists of splitting the box-constraints domain into smaller subdomains in order to reduce the gap between an upper and a lower bound on the minimum of the problem.The upper bound is given by the smallest functional value obtained through local minimizations within the subdomains,while the lower bound comes from the global minimization of the convexα-underestimators subject to the problem constraints.If, within a subdomain,the lower bound plus the prescribed toleranceεk is above the upper bound, the subdomain can be discarded as it clearly does not contain the solution of the problem (considering the toleranceεk).The same argument applies if,via interval analysis,it is shown the subdomain does not contain any promising point.

The constraints of the problem,or any other valid constraint,can also be used to substitute a subdomain[ˉl,ˉu]for any other proper subdomain[?l,?u].In our implementation,we considered just linear constraints in the process of reducing a subdomain[ˉl,ˉu].Three sources of linear constraints were used,namely(a)linear constraints of the original problem;(b)linear relaxations (valid within the subdomain[ˉl,ˉu])of the nonlinear penalized constraints;and(c)linear“cuts”of

the form L Uρ

k (x,λk,μk)≤L ub,where L Uρ

(·,λk,μk)is a linear relaxation of Lρ

(·,λk,μk)within

[ˉl,ˉu].Constraints of type(a)and(b)can be used to discard regions that do not contain feasible

points of problem(1)and play a role in the de?nition of P k.Constraints of type(c)eliminate

regions that do not contain the global solution of(8).

Let us call B the original box of the problem and L the original polytope de?ned by the

linear constraints.As a result of theαBB process,the original box is covered by t“small”

boxes B1,...,B t.For each small box B i a polytope Q i(de?ned by the relaxations)is given

(perhaps Q i=I R n)and a new small box?B i such that(Q i∩B i)??B i?B i is constructed.By construction,the set P k=∪t i=1?B i contains the feasible set of the problem.So,P k contains the

global minimizers of the problem.TheαBB algorithm guarantees anεk-global minimizer on L∩P k,as required by the theory.

The algorithm starts with a list S of unexplored subdomains that initially has as unique element the original box domain of the problem.Then,for each subdomain in the list it does the following tasks:(i)reduce the subdomain;(ii)try to discard the subdomain via interval analysis or computing the global solution of the underestimating problem;(iii)if the subdomain cannot be discarded;perform a local minimization within the subdomain;(iv)?nally,split the subdomain and add the new subdomains to S.The method stops when the list S is empty.

The description of theαBB algorithm,that follows very closely the algorithm introduced in [3],is as follows.

Algorithm4.1:αBB

Step1.Initialization

Set S={[l,u]}and L ub=+∞.

Step2.Stopping criterion

If S is empty,stop.

Step3.Choose a subdomain

Choose[ˉl,ˉu]∈S and set S←S\[ˉl,ˉu].

Step4.Subdomain reduction and possible discarding

Let W[ˉl,ˉu]be a set of linear constraints valid within the subdomain[ˉl,ˉu]plus linear con-straints satis?ed by the global solution of(8).For i=1,...,n,compute?l i and?u i as

arg min±x i subject to Ax=b,Cx≤d,ˉl≤x≤ˉu,x∈W[ˉl,ˉu],(9) respectively.If the feasible set of(9)is empty,discard the subdomain[ˉl,ˉu]and go to Step2.

Step5.Reduced subdomain discarding

https://www.wendangku.net/doc/0b17638828.html,ing interval analysis,compute[L min

[?l,?u],L max

[?l,?u]

]such that

L min

[?l,?u]≤Lρk(x,λk,μk)≤L max[?l,?u],?x∈[

?l,?u].

If L min

[?l,?u]

+εk≥L ub then discard the reduced subdomain[?l,?u]and go to Step2.

https://www.wendangku.net/doc/0b17638828.html,pute a convex underestimator U

[?l,?u](x)of Lρ

(x,λk,μk)and?nd

y1←arg min U[?l,?u](x)subject to Ax=b,Cx≤d,?l≤x≤?u.

If U

[?l,?u]

(y1)+εk≥L ub then discard the reduced subdomain[?l,?u]and go to Step2.

Step6.Local optimization within subdomain

Using y1as initial guess,compute

y2←arg min Lρk(x,λk,μk)subject to Ax=b,Cx≤d,?l≤x≤?u.

(y2,λk,ρk)

If Lρ

Step7.Split subdomain

Split[?l,?u]in at least2proper subdomains.Add the new subdomains to S and go to Step3.

Remarks.

1.The set S is implemented as a queue(see Steps3and7).It is a very simple and problem-

independent strategy that,in contrast to the possibility of using a stack,provides to the method a diversi?cation that could result in fast improvements of L ub.

2.At Step7,we divide the subdomain in two subdomains splitting the range of the variable

selected by the least reduced axis rule[3]in its middle https://www.wendangku.net/doc/0b17638828.html,ly,we choose x i such that i=arg min j{(?u j??l j)/(u j?l j)}.If a variable appears linearly in the objective function of the subproblem(8)then its range does not need to be split at all and it is excluded from the least reduced axis rule.The variables that appear linearly in the Augmented Lagrangian are the ones that appear linearly in the objective function of the original problem(1)and do not appear in the penalized nonlinear constraints.

3.Set W[ˉl,ˉu]at Step4is composed by linear relaxations(valid within the subdomain[ˉl,ˉu])

of the penalized nonlinear constraints.We considered the convex and concave envelopes for bilinear terms and the convex envelope for concave functions[15],as well as tangent hyperplanes to the convex constraints.

4.By the de?nition of the Augmented Lagrangian function(2),it is easy to see that f(x)≤

Lρ(x,λ,μ)?x.So,any linear relaxation of the objective function f(x)is also a linear relaxation of the Augmented Lagrangian function Lρ(x,λ,μ).Therefore,a constraint of the form f U(x)≤L ub,where f U(·)is a linear relaxation of f(·)for all x∈[ˉl,ˉu]was also included in the de?nition of W[ˉl,ˉu]at Step4.

5.As a consequence of the de?nition of?in(8)(associated to the choice of the lower-level

constraints),the optimization subproblems at Steps5.2and6are linearly constrained op-timization problems.Moreover,by the de?nition of?and the fact that W[ˉl,ˉu]at Step4 is described by linear constraints,the2n optimization problems at Step4are linear pro-gramming problems.

6.At Step5.2,it is not necessary to complete the minimization process.It would be enough

to?nd y such that U

(z)+εk

[?l,?u]

There is an alternative that represents a trade-o?between e?ectiveness and e?ciency.Step 4can be repeated,substituting ˉl and ˉu by ?l and ?u in (9),respectively,while ?l =ˉl or ?u =ˉu .Moreover,within the same loop,ˉl i and ˉu i can be replaced by ?l i and ?u i as soon as they are computed.Doing that,the order in which the new bounds are computed might in?uence the ?nal result,and a strategy to select the optimal sequence of bounds updates might be developed.

The computation of the convex underestimator U [?l,?u ](x )uses the convex α-underestimator for general nonconvex terms introduced in [31]and de?ned as follows:

U [?l,?u ](x )=L ρk (x,λk ,μk

n i =1

αi (?u ?x )(x ??l ),

where αi ,i =1,...,n ,are positive scalars large enough to ensure the convexity of U [?l,?u ](x ).

We used the Scaled Gerschgorin method [2,3]to compute the α’s.Given d ∈I R ++and

[H [?l,?u ]],the interval Hessian of L ρk (x,λk ,μk )for the interval [?l,?u

],we have αi =max {0,?

d i

)},i =1,...,n,

(10)

where |h |ij =max {|h min ij |,|h max ij |}and (h min ij ,h max ij )denotes element at position (i,j )of [H [?l,?u ]].

As suggested in [2],we choose d =?u ??l .

The piecewise convex α-underestimator [32]can also be used instead of the convex α-underestimator to compute the Augmented Lagrangian underestimator at Step 5.2.The latter one may be signi?cantly tighter than the ?rst one,while its computation is more time consum-ing.It is worth mentioning that,although the theory of both underestimators was developed for twice-continuously di?erentiable functions,the continuity of the second derivatives,which does not hold in the Augmented Lagrangian (2),is not necessary at all.The computation of the piecewise convex α-underestimator deserves further explanations.

Considering,for each interval [?l i ,?u i ],i =1,...,n ,a partitioning in N i subintervals with end-points ?l i ≡?v 0i ,?v 1i ,...,?v N i i ≡ˉu i ,a de?nition of the piecewise convex α-underestimator,equivalent to the one introduced in [32],follows:

Φ[?l,?u ]

(x )=L ρk (x,λk ,μk )? n

i =1q i (x ),q i (x )

=q j i (x ),if x ∈[?v j ?1i ,?v j

i ],i =1,...,n,j =1,...,N i ,

q j

i (x )

=αj i (?

v j i ?x i )(x i ??v j ?1i )+βj i x i +γj

i ,i =1,...,n,j =1,...,N i .In the de?nition above,α’s,β’s and γ’s are such that q i (x ),i =1,...,n ,are continuous and

smooth and Φ[?l,?u ](x )is a convex underestimator of L ρk (x,λk ,μk )within the box [?l,?u ].One way to compute the α’s is to compute one α∈I R n for each one of the

n i =1N i subdo-mains and set αj

i ,i =1,...,n,j =1,...,N i ,as the maximum over all the [α]i ’s of the subdo-mains included in the “slice”[(?l 1,...,?l i ?1,?v j ?1i ,?l i +1,...,?l n ),(?u 1,...,?u i ?1,?v j i ,?u i +1,...,?u n )].In this way, n

i =1N i α’s must be computed.

After having computed the α’s,and considering that each interval [?l i ,?u i ]is partitioned in N i identical subintervals of size s i =(?u i ??l i )/N i ,β’s and γ’s can be computed as follows:for each

i=1,...,n,

β1i=?s i

h(x0) 2+ g(x0)+ 2

where x0is an arbitrary initial point.As stopping criterion we used max( h(x k) ∞, V k ∞)≤10?4.Several tests were done in order to analise the behaviour of the method in relation to the choice of the optimality gapsεk for theεk-global solution of the subproblems andεfor the ε-global solution of the original problem.All the experiments were run on a3.2GHz Intel(R) Pentium(R)with4processors,1Gb of RAM and Linux Operating https://www.wendangku.net/doc/0b17638828.html,piler option “-O”was adopted.

We selected a set of18problems from the literature(see Appendix).In a?rst experiment,we solved the problems using the convexα-underestimator and using a variableεk=max{ε,10?k} and a?xedεk=ε.In both cases we consideredε∈{10?1,10?2,10?3,10?4}.Table1shows the results.In Table1,n is the number of variables and m is the number of constraints.The number within parentheses is the number of linear constraints.“Time”is the total CPU time in seconds,“It”is the number of iterations of Algorithm2.1(outer iterations)which is equal to the number of subproblems(8)that are solved toεk-global optimality using theαBB approach, and“#Nodes”is the total number of iterations of Algorithm4.1(inner iterations).The method with the eight combinations ofεk andεfound the same global minimum in all the problems.As expected,smaller gaps require more computations.However,the e?ort increase is not the same in all the problems,as it strongly depends on the tightness of the convexα-underestimator in relation to the Augmented Lagrangian function and the di?culty in closing that gap for each particular problem.

In a second experiment,we setεk=max{ε,10?k}andε=10?4(which provides the tightest global optimality certi?cation)and compare the performance of the method using the convex α-underestimator and the piecewise convexα-underestimator with N i≡N≡2?i.Table2 shows the results.In Table2,f(x?)is the global minimum.As expected,both versions of the method found the global minimum reported in the literature up to the prescribed tolerances. The version that uses the convexα-underestimator required less CPU time while the version that used the piecewise convexα-underestimator used a smaller number of inner iterations.

It is well known that the performance of a global optimization method for solving a problem strongly depends on the particular way the problem is modeled.So,it is worth mentioning that all the test problems,apart from Problems2(a–d),were solved using the models shown in the Appendix without any kind of algebraic manipulation or reformulation.One of the key problem features that a?ects the behaviour of the present approach is the number of variables that appear nonlinearly in the objective function or in a nonlinear constraint.Those are the variables that need to be branched in theαBB method.In the formulation of Problems2(a–c) presented in the Appendix,all the nine variables need to be branched.As a result,the direct application of Algorithm2.1to those problems takes several seconds of CPU time.On the other hand,the addition of new variables w78=x7x8and w79=x7x9reduces the number of variables that need to be branched from nine to?ve and the CPU time used by the method to a fraction of a second.Almost identical reasoning applies to Problem2(d).

Finally,note that,as Problems11and16have just lower-level constraints,the Augmented Lagrangian framework is not activated at all and its resolution is automatically made through the direct application of theαBB method.

These numerical results are the?rst step towards corroboration of the practical reliability of our method.The Augmented Lagrangian ideas are di?erent from the ones that support current available global nonlinear programming solvers[34],in particular,the ones involved in the broad numerical study[35].However,most of the interest of the Augmented Lagrangian approach relies on the fact that one is not restricted to the use of a particular linear-constraint solver.The Augmented Lagrangian method may use linear-constraint solvers quite di?erent from αBB,if this is required by particular characteristics of the problem.Moreover,if the constraints of a particular problem can be divided into two sets,Easy and Di?cult,and problems with Easy constraints(not necessarily linear)may be e?ciently solved by some other solver,an appealing

Variableεk=max{ε,10?k}

Time It#Nodes Time It#Nodes 59.4495986115.839103919

110.098780.138100

110.64135630.7313651

110.078600.17884

120.042200.14363

6 1.18614767.78612688

20.1921294 1.3938719

20.002680.013129

100.00130.0013

20.002220.00341

30.0377650.037817

20.0153430.015467

20.0131490.013283

20.10623400.1463496

60.00130.0013

20.003950.013139

20.002220.003105

20.0182400.018334

30.1088660.4383750

40.00110.0011

30.011990.033565

30.001230.00375

50.0261160.126698

Fixedεk=ε

Time It#Nodes Time It#Nodes 59.4695986115.889103919

110.098780.128100

110.64135630.7313651

110.078600.17884

120.042200.08238

6 1.17614767.79612694

20.1921294 1.5229580

20.012680.01298

100.00130.0013

20.002220.00234

30.0477650.037819

20.0153430.015515

20.0131490.023403

20.10623400.1563692

60.00150.0015

20.003950.013181

20.002220.00278

20.0182400.018342

30.108866 1.21810608

40.00110.0011

30.011990.021299

30.001230.00127

50.0261160.116698

Problem m αBB PiecewiseαBB(N=2)

Time It#Nodes

1342.579117679

110.138104?4.0000E+02 2(b)8(6) 2.2313671

110.16888?7.5000E+02 2(d)9(7) 3.444110

612.07620366?3.8880E?01 3(b)1 5.62413954

20.014182?9.4773E+00 5100.0113

20.00460?6.6666E+00 730.047561

20.015493 3.7629E+02 94(2)0.024372

20.1563918?1.1870E+02 116(6)0.0115

20.0142067.4178E?01 1310.014160

20.018370?1.6739E+01 152(1) 1.4783820

40.0011?4.5142E+00 17(a)3(1)0.1841558

30.0141080.0000E+00 183(1)0.226758

idea is to de?ne the Augmented Lagrangian only with respect to the Di?cult constraints and employ the other solver for the subproblems.This is the approach of[7]for constrained local optimization.External Penalty methods share this characteristic of the Augmented Lagrangian algorithm,but they tend to produce very di?cult subproblems for big penalty parameters.In the Augmented Lagrangian method,the penalty parameter tends to remain bounded[7].

6Final remarks

As already mentioned in[7],one of the advantages of the Augmented Lagrangian approach for solving nonlinear programming problems is its intrinsic adaptability to the global optimization https://www.wendangku.net/doc/0b17638828.html,ly,if one knows how to solve globally simple subproblems,the Augmented La-grangian methodology allows one to globally solve the original constrained optimization problem. In this paper,we proved rigorously this fact,with an additional improvement of the Augmented Lagrangian method:the subproblems are rede?ned at each iteration using an auxiliary constraint set P k that incorporates information obtained on the?ight about the https://www.wendangku.net/doc/0b17638828.html,ing theαBB algorithm for linearly constrained minimization subproblems,we showed that this approach is reliable.As a result,we have a practical Augmented Lagrangian method for constrained op-timization that provably obtainsε-global minimizers.Moreover,the Augmented Lagrangian approach has a modular structure thanks to which one may easily replace subproblem global optimization solvers.This means that our method will be automatically improved as long as new global optimization solvers will be developed for the simple subproblems.

The challenge is improving e?ciency.There are lots of unconstrained and constrained global optimization problems in Engineering,Physics,Chemistry,Economy,Computational Geometry and other areas that are not solvable with the present computer facilities.Our experiments seem to indicate that there is little to improve in the Augmented Lagrangian methodology,since the number of outer iterations is always moderate.The?eld for improvement is all concentrated in the global optimization of the subproblems.So,much research is expected in the following years in order to be able to e?ciently solve more challenging practical problems.

TheαBB global optimization algorithm is fully parallelizable in at least two levels:(i) subproblems with di?erent subdomains can be solved in parallel;and(ii)the piecewise convex α-underestimator can be computed in parallel,reducing the number of nodes in the Branch& Bound algorithm without increasing the CPU time.Also the development of tighter underesti-mators for general nonconvex terms would be the subject of further research. References

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7Appendix

In this Appendix we describe the global optimization test problems considered in the numerical experiments.

Problem1.[33]

Minimize(x1?1)2+(x1?x2)2+(x2?x3)3+(x3?x4)4+(x4?x5)4

√

subject to x1+x22+x33=3

2?2

x1x5=2

?5≤x i≤5,i=1,...,5

Problem2.Haverly’s pooling problem[3].

Minimize?9x1?15x2+6x3+c1x4+10(x5+x6)

subject to x7x8+2x5?2.5x1≤0

x7x9+2x6?1.5x2≤0

3x3+x4?x7(x8+x9)=0

x8+x9?x3?x4=0

x1?x8?x5=0

x2?x9?x6=0

(0,...,0)≤x≤(c2,200,500, (500)

(a)c1=16and c2=100;(b)c1=16and c2=600;(c)c1=13and c2=100. Problem2(d).A very similar version of the problem above but with an additional variable and di?erent bounds[40].

Minimize?9x5?15x9+6x1+16x2+10x6

subject to x10x3+2x7?2.5x5≤0

x10x4+2x8?1.5x9≤0

3x1+x2?x10(x3+x4)=0

x1+x2?x3?x4=0

x3+x7?x5=0

x4+x8?x9=0

x7+x8?x6=0

(0,...,0,1)≤x≤(300,300,100,200,100,300,100,200,200,3) Problem3.Reactor network design[30].

Minimize?x4

subject to x1+k1x1x5=1

x2?x1+k2x2x6=0

x3+x1+k3x3x5=1

x4?x3+x2?x1+k4x4x6=0

√x

6≤4

(0,0,0,0,10?5,10?5)≤x≤(1,1,1,1,16,16) k1=9.75598810?2;k2=0.99k1;k3=3.91908010?2;k4=0.90k3.

煤的各项指标代码及意义

————————————————————————————————作者: ————————————————————————————————日期： ?

煤的各项指标代码及意义 1、水分(代码Ｍ或Ｗ) 煤的水分是指单位质量的煤中水的含量。煤的水分有内在水分和外在水分两种，两者之和为全水分(Mｔ ); 进行煤的工业分析所测定煤的水分为空气干燥煤样的水分(Mad)。煤的水分是评价煤炭经济价值的基本指标。煤的内在水分与煤的煤化程度和内部表面积有关，一般来说变质程度越低,煤的内表面积越大，水分含量越高，经济价值越低。煤的水分对其存储、运输、加工和利用均有影响。在存储时,水分能加速煤的风化、碎裂、自燃;在运输中,水分会增加运输量,加大运费,并会增加装车、卸车的困难。在寒冷地区,水分大的煤在长途运输中会冻结,给卸车造成极大困难。煤的水分在燃烧时要消耗一定的热量,在炼焦时要延长结焦时间,而且影响焦煤的寿命。

2、灰分(代码A) 煤的灰分是指煤完全燃烧后残留物的产率。煤的灰分分为内在灰分和外在灰分两种。内在灰分是指煤在成煤过程中混入的矿物杂质;外在灰分是指煤在开采、运输、储存过程中混入的矿物杂质，即矸石,可以通过洗选方法出去。煤的灰分是衡量煤炭质量的一个重要指标,灰分越高，质量就越差，发热量就越低。煤的灰分对煤的加工利用有不利影响。外在灰分越高,在洗选时排除的矸石量越大；内在灰分越高，煤就越难选。煤的灰分高，会增加运输量和运费。在燃烧时,灰分越高，热效率越低，而且会增加烟尘排放量和炉渣量，加剧燃煤对大气的污染。炼焦时，精煤灰分越高,焦炭的灰分就越高，炼铁的焦比就增加，高炉利用系数就越低，产铁量减少。

unit4globalwarming单词和句型重点总结

Unit 4 Global warming全球变暖一、词汇 about发生；造成注意：（1）come about是不及物动词短语，不能用于被动语态，常指情况不受人控制的突然发生。有时用it作形式主语，that从句作真正主语。（2）表示“发生”的词或短语有：happen,occur,take place,break ① Many a quarrel comes about through a misunderstanding. ② The moon came out from behind the clouds. ③ I’ll let you know if anything comes up. ④ I’ll come over and see how you are coming along. ⑤ I came across an old friend yesterday. ⑥ When she came to, she couldn’t recognize the surroundings. ① I subscribe to your suggestion. ② Which magazine do you subscribe to? ③ He subscribed his name to the paper（文件）. ④ He subscribed a large sum to the poor students. n.量；数量

① It’s cheaper to buy goods in quantity / in large quantities. ② A large quantity of silk is sold in Japan. ③ A large quantity of drugs are found in his home. ④ Large quantities of rain are needed in this area. ① He tends to get angry when others disagree with her. ② His views tend towards the extreme（极端）. ③ He was tending (to) his son when I saw him in the hospital. ④ Jane is nice but has a tendency to talk too much. =Jane is nice but she tends to talk too much. ① The price of the new house in our area has gone up by 1,000 yuan per square meter（平方米）。That is（也就是说）it has gone up to 5,000 yuan per square meter. ② The wind has gone down a little. ④ The country has gone through too many wars.

煤炭指标及煤种

焦炭：烟煤在隔绝空气的条件下，加热到950-1050℃，经过干燥、热解、熔融、粘结、固化、收缩等阶段最终制成焦炭，这一过程叫高温炼焦（高温干馏）。精煤：原煤经过洗煤，除去煤炭中矸石，即为精煤。肥煤是指国家煤炭分类标准中，对煤化变质中等，粘结性极强的烟煤的称谓，炼焦煤的一种，炼焦配煤的重要组成部分，结焦性最强，熔融性好，结焦膨胀度大，耐磨；精煤是指经洗选加工供炼焦用或其他用途的洗选煤炭产品的总称。煤的挥发分煤的挥发分，即煤在一定温度下隔绝空气加热，逸出物质（气体或液体）中减掉水分后的含量。剩下的残渣叫做焦渣。因为挥发分不是煤中固有的，而是在特定温度下热解的产物，所以确切的说应称为挥发分产率。（1）煤的挥发分不仅是炼焦、气化要考虑的一个指标，也是动力用煤的一个重要指标，是动力煤按发热量计价的一个辅助指标。挥发分是煤分类的重要指标。煤的挥发分反映了煤的变质程度，挥发分由大到小，煤的变质程度由小到大。如泥炭的挥发分高达70%，褐煤一般为40～60%，烟煤一般为10～50%，高变质的无烟煤则小于10%。煤的挥发分和煤岩组成有关，角质类的挥发分最高，镜煤、亮煤次之，丝碳最低。所以世界各国和我国都以煤的挥发分作为煤分类的最重要的指标。（2）煤的挥发分测试。从广义上来讲，凡是以发电、机车推进、锅炉燃烧等为目的，产生动力而使用的煤炭都属于动力用煤，简称动力煤。 1)无烟煤(WY)。无烟煤固定碳含量高，挥发分产率低，密度大，硬度大，燃点高，燃烧时不冒烟。01号无烟煤为年老无烟煤；02号无烟煤为典型无烟煤；03号无烟煤为年轻无烟煤。如北京、晋城、阳泉分别为01、02、03号无烟煤。 2)贫煤(PM)。贫煤是煤化度最高的一种烟煤，不粘结或微具粘结性。在层状炼焦炉中

煤炭分类及标准

中国煤炭分类国家标准表判别煤炭质量优劣的指标很多，其中最主要的指标为煤的灰分含量和硫分含量。一般陆相沉积，煤的灰分、硫分普遍较低；海陆相交替沉积，煤的灰分、硫分普遍较高。中国煤炭灰分普遍较高，秦岭以北地区，晋北、陕北、宁夏、两淮、东北等地区，侏罗纪煤田为陆相沉积，煤的灰分一般为 10％～20％，有的在10％以下，硫分一般小于1％，东北地区硫分普遍小于0．5％。中国北方普遍分布的石灰纪、秦岭以南地区、湖南的黔阳煤系、湖北的梁山煤系等属海陆交替沉积的煤，灰分一般达15％～25％，硫分一般高达2％～5％。广西合山、四川上寺等地的晚二叠纪煤层属浅海相沉积煤，硫分可高达6％～10％以上。据统计，中国灰分小于10％的特低灰煤仅占探明储量的17％左右。大部分煤炭的灰分为10％～30％。硫分小于1％的特低硫煤占探明储量的43．5％以上，大于4％的高硫

煤仅为2．28％。中国的炼焦用煤一般为中灰、中疏煤，低灰和低硫煤很少。炼焦用煤的灰分一般都在20％以上；硫分含量大于2％的炼焦用煤占20％以上。中国炼焦用煤的另一大特点是：硫分越高，煤的动结性往往越强，其可选性一般较差。中国褐煤多属老年褐煤。褐煤灰分一般为20％～30％。东北地区褐煤硫分多在1％以下，广东、广西、云南褐煤硫分相对较高，有的甚至高达8％以上。褐煤全水分一般可达20％～50％，分析基水分为10％~20％，低位发热量一般只有11．71～16．73MJ／kg。中国烟煤的最大特点是低灰、低硫；原煤灰分大都低于15％，硫分小于1％。部分煤田，如神府、东胜煤田，原煤灰分仅为3％一5％，被誉为天然精煤。烟煤的第二个特点是煤岩组分中丝质组含量高，一般在40％以上，因此中国烟煤大多为优质动力煤。中国贫煤的灰分和硫分都较高，其灰分大多为15％-30％，流分在1．5％-5％之间。贫煤经洗选后，可作为很好的动力煤和气化用煤。中国典型的无烟煤和老年无烟煤较少，大多为三号年轻无烟煤，其主要特点是，灰分和硫分均较高，大多为中灰、中硫、中等发热量、高灰熔点，主要用作动力用煤，部分可作气化原料煤。

煤炭的各项指标

煤炭的各项指标第一个指标：水分。煤中水分分为内在水分、外在水分、结晶水和分解水。煤中水分过大是，不利于加工、运输等，燃烧时会影响热稳定性和热传导，炼焦时会降低焦产率和延长焦化周期。现在我们常报的水份指标有： 1、全水份（Mt），是煤中所有内在水份和外在水份的总和，也常用Mar表示。通常规定在8%以下。 2、空气干燥基水份(Mad)，指煤炭在空气干燥状态下所含的水份。也可以认为是内在水份，老的国家标准上有称之为“分析基水份”的。第二个指标：灰分指煤在燃烧的后留下的残渣。不是煤中矿物质总和，而是这些矿物质在化学和分解后的残余物。灰分高，说明煤中可燃成份较低。发热量就低。同时在精煤炼焦中，灰分高低决定焦炭的灰分。能常的灰分指标有空气干燥基灰分（Aad）、干燥基灰分（Ad）等。也有用收到基灰分的（Aar）。第三指标：挥发份（全称为挥发份产率）V 指煤中有机物和部分矿物质加热分解后的产物，不全是煤中固有成分，还有部分是热解产物，所以称挥发份产率。挥发份大小与煤的变质程度有关，煤炭变质量程度越高，挥发份产率就越低。在燃烧中，用来确定锅炉的型号；在炼焦中，用来确定配煤的比例；同时更是汽化和液化的重要指标。常使用的有空气干燥基挥发份（Vad）、干燥基挥发份（Vd）、干燥无灰基挥发份（Vdaf）和收到基挥发份（Var）。其中Vdaf是煤炭分类的重要指标之一。其他指标：煤炭的固定碳（FC）固定碳含量是指去除水分、灰分和挥发分之后的残留物，它是确定煤炭用途的重要指标。从100减去煤的水分、灰分和挥发分后的差值即为煤的固定碳含量。根据使用的计算挥发分的基准，可以计算出干基、干燥无灰基等不同基准的固定碳含量。发热量（Q）发热量是指单位质量的煤完全燃烧时所产生的热量，主要分为高位发热量和低位发热量。煤的高位发热量减去水的汽化热即是低位发热量。发热量的国标单位为百万焦耳/千克（MJ/KG）常用单位大卡/千克，换算关系为：1MJ/KG=239.14Kcal/kg;1J=0.239cal;1cal=4.18J。如发

高中英语《Unit4Globalwarming》课文语法填空新人教版

《Unit 4 Global warming》 Does It Matter? 一、语法填空(根据课文内容、依据语法规则完成下面短文） When 1________ (compare) with most natural changes, that the temperature of the earth rose about one degree Fahrenheit during the 20th century is quite shocking. And it’s human activity 2________ has caused this global warming rather 3________ a random but natural phenomenon. Dr. Janice Foster explains that we add huge 4________ of extra carbon dioxide into the atmosphere by burning fossil 5________. From the second 6________ and the discovery of Charles Keeling, all scientists believe that the burning of more and more fossil fuels 7________ (result) in the increase in carbon dioxide. Greenhouse gases continue to build up. On the one hand, Dr. Foster thinks that the trend would be a 8________. On the other hand, George Hambley 9________ (state) that more carbon dioxide would encourage a greater range of animals and bring us 10________ better life.

煤炭指标-煤炭的各项指标-六大指标

煤炭指标-煤炭的各项指标-六大指标来源:中国煤炭价格网数据整理煤炭六项基本指标：第一个指标：水分。煤中水分分为内在水分、外在水分、结晶水和分解水。煤中水分过大是，不利于加工、运输等，燃烧时会影响热稳定性和热传导，炼焦时会降低焦产率和延长焦化周期。现在我们常报的水份指标有： 1、全水份（Mt），是煤中所有内在水份和外在水份的总和，也常用Mar表示。通常规定在8%以下。 2、空气干燥基水份(Mad)，指煤炭在空气干燥状态下所含的水份。也可以认为是内在水份，老的国家标准上有称之为“分析基水份”的。第二个指标：灰分指煤在燃烧的后留下的残渣。不是煤中矿物质总和，而是这些矿物质在化学和分解后的残余物。灰分高，说明煤中可燃成份较低。发热量就低。同时在精煤炼焦中，灰分高低决定焦炭的灰分。能常的灰分指标有空气干燥基灰分（Aad）、干燥基灰分（Ad）等。也有用收到基灰分的（Aar）。

第三指标：挥发份（全称为挥发份产率）V 指煤中有机物和部分矿物质加热分解后的产物，不全是煤中固有成分，还有部分是热解产物，所以称挥发份产率。挥发份大小与煤的变质程度有关，煤炭变质量程度越高，挥发份产率就越低。在燃烧中，用来确定锅炉的型号；在炼焦中，用来确定配煤的比例；同时更是汽化和液化的重要指标。常使用的有空气干燥基挥发份（Vad）、干燥基挥发份（Vd）、干燥无灰基挥发份（Vdaf）和收到基挥发份（Var）。其中Vdaf是煤炭分类的重要指标之一。第四个指标：固定碳不同于元素分析的碳，是根据水分、灰分和挥发份计算出来的。 FC+A+V+M=100 相关公式如下：FCad=100-Mad-Aad-Vad FCd=100-Ad-Vd FCdaf=100-Vdaf 第五个指标：全硫St 是煤中的有害元素，包括有机硫、无机硫。1%以下才可用于燃料。部分地区要求在和以下，现在常说的环保煤、绿色能源均指硫份较低的煤。常用指标有：空气干燥基全硫(St,ad）、干燥基全硫及收到基全硫(St,ar)。

中国煤炭分类、煤质指标的分级

煤质指标的分级中国煤炭分类 (2008-06-19 10:04:30)

??中国煤炭分类：首先按煤的挥发分，将所有煤分为褐煤、烟煤和无烟煤；对于褐煤和无烟煤，再分别按其煤化程度和工业利用的特点分为2个和3个小类；烟煤部分按挥发分>10%~20%、>20%~28%、28%~37和>37%的四个阶段分为低、中、中高及高挥发分烟煤。关于烟煤粘结性，则按粘结指数G区分：0~5为不粘结和微粘结煤；>5~20为弱粘结煤；>20~50为中等偏弱粘结煤；>50~65为中等偏强粘结煤；>65则为强粘结煤。对于强粘结煤，又把其中胶质层最大厚度Y>25mm或奥亚膨胀度b>150%（对于Vdaf>28%的烟煤，b>220%）的煤分为特强粘结煤。在煤类的命名上，考虑到新旧分类的延续性，仍保留气煤、肥煤、焦煤、瘦煤、贫煤、弱粘煤、不粘煤和长焰煤8个煤类。 ????在烟煤类中，对G>85的煤需再测定胶质层最大厚度Y值或奥亚膨胀度B值来区分肥煤、气肥煤与其它烟煤类的界限。当Y值大于25mm时，如Vdaf>37%，则划分为气肥煤。如Vdaf<37%，则划分为肥煤。如Y值<25mm，则按其Vdaf值的大小而划分为相应的其它煤类。如Vdaf>37%，则应划分为气煤类，如Vdaf>28%-37%，则应划分为1/3焦煤，如Vdaf在于28%以下，则应划分为焦煤类。 ????这里需要指出的是，对G值大于100的煤来说，尤其是矿井或煤层若干样品的平均G值在100以上时，则一般可不测Y值而确定为肥煤或气肥煤类。 ????在我国的煤类分类国标中还规定，对G值大于85的烟煤，如果不测Y值，也可用奥亚膨胀度B值（%）来确定肥煤、气煤与其它煤类的界限，即对Vdaf<28%的煤，暂定b值>150%的为肥煤；对Vdaf>28%的煤，暂定b值>220%的为肥煤（当Vdaf值<37%时）或气肥煤（当Vdaf值>37%时）。当按b值划分的煤类与按Y值划分的煤类有矛盾时，则以Y值确定的煤类为准。因而在确定新分类的强粘结性煤的牌号时，可只测Y值而暂不测b值。（中国煤煤分类国家标准表）

煤炭的分类和用途

煤炭的种类及用途煤炭的分类：煤主要有褐煤、烟煤、无烟煤、半无烟煤等几种。（1）褐煤：多为块状，呈黑褐色，光泽暗，质地疏松；含挥发分40%左右，燃点低，容易着火，燃烧时上火快，火焰大，冒黑烟；含碳量与发热量较低（因产地煤级不同，发热量差异很大），燃烧时间短，需经常加煤。（2）烟煤：一般为粒状、小块状，也有粉状的，多呈黑色而有光泽，质地细致，含挥发分30%以上，燃点不太高，较易点燃；含碳量与发热量较高，燃烧时上火快，火焰长，有大量黑烟，燃烧时间较长；大多数烟煤有粘性，燃烧时易结渣。（3）无烟煤：有粉状和小块状两种，呈黑色有金属光泽而发亮。杂质少，质地紧密，固定碳含量高，可达80%以上；挥发分含量低，在10%以下，燃点高，不易着火；但发热量高，刚燃烧时上火慢，火上来后比较大，火力强，火焰短，冒烟少，燃烧时间长，粘结性弱，燃烧时不易结渣。应掺入适量煤土烧用，以减轻火力强度。 1989年10月，国家标准局发布《中国煤炭分类国家标准》(GB5751-86)，依据干燥无灰基挥发分Vdaf、粘结指数G、胶质层最大厚度Y、奥亚膨胀度b、煤样透光性P、煤的恒湿无灰基高位发热量Qgr，maf等6项分类指标，将煤分为14类。即褐煤、长焰煤、不粘煤、弱粘煤、1／2中粘煤、气煤、气肥煤、1／3焦煤、肥煤、焦煤、瘦煤、贫瘦煤、贫煤和无烟煤。 1．无烟煤：高固定碳含量，高着火点（约360～420℃）,高真相对密度（1.35～1.90）,低挥发分产量和低氢含量。除了发电外，无烟煤主要作为气化原料（固定床气化发生炉）用于合成氨、民用燃料及型煤的生产等。一些低灰低硫高HGI的无烟煤也用于高炉喷吹的原料。2．贫煤：煤烟中煤级最高的煤，它的特征是：较高的着火点（350—360℃），高发热量，弱粘结性或不粘结。贫煤主要用于发电和电站锅炉燃料。使用贫煤时，将其与其他一些高挥发分煤配合使用也不失为一个好的途径。 3．贫瘦煤：挥发分低，粘结性较差，可以单独用来炼焦。当与其他适合炼焦的煤种混合时，贫瘦煤的掺入将使焦炭产品的块度增大。贫瘦煤也可用于发电、电站锅炉和民用燃料等方面。典型的贫瘦煤产于山西省西山煤电公司。 4．瘦煤：中度的挥发分和粘结性，主要用于炼焦。在炼焦过程中可能会产生一些胶质物，胶质层的厚度为6—10mm。由瘦煤单独炼焦产生的焦炭，机械强度较高但耐磨强度相对较差。除了那部分高灰高硫的瘦煤，瘦煤经常与其他煤种混合炼焦。 5．焦煤：有很强的炼焦性，中等的挥发分（约16％—28％），焦煤是国内主要用于炼焦的煤种。由焦煤炼成的焦炭具有非常优良的性质，焦煤主要产于山西省和河北省。 6．肥煤：中等或较高的挥发分（约25％—35％）和很强的粘结性，主要用于炼焦（一些高灰高硫的肥煤用来发电）。与其他煤级的煤相比，肥煤一般具有较高的硫含量。 7．1/3焦煤：介于焦煤、气煤和肥煤之间，具有较高的挥发分（类似于气煤），较强的粘结性（类似于肥煤）和很好的炼焦性（类似于焦煤），这也是它被称为1/3焦煤的原因。1/3焦煤由于其产量高而主要用于炼焦和发电。 8．气肥煤：高挥发分（接近于气煤）和强的粘结性（接近于肥煤），它适用于焦化作用产生的城市燃气和与其他煤种混合炼焦以增加煤气、焦油等副产品的产量。气肥煤的显微组成与其他煤种有很大的差异，壳质组的含量相对较高。 9．气煤：很高的挥发分和中度的粘结性，主要用于炼焦和发电。典型的气煤产于辽宁省。

Global warming全球变暖全英文介绍

One of the effects of global warming is the destruction of many important ecosystems.Changing and erratic climate conditions will put our ecosystems to the test, the increase in carbon dioxide will increase the problem. The evidence is clear that global warming and climate change affects physical and biological systems. There will be effects to land, water, and life. Already today, scientists are seeing the effects of global warming on coral reefs, many have been bleached and have died. This is due to warmer ocean waters, and to the fact that some species of plants and animals are simply migrating to better suited geographical regions where water temperatures are more suitable. Melting ice sheets are also making some animals migrate to better regions. This effects the ecosystems in which these plants and animals live. Several climate models have been made and they predict more floods (big floods), drought, wildfires, ocean acidification, and the eventual collapse of many ecosystems throughout the world both on land and at sea. There have been forecasts of things like famine, war, and social unrest, in our days ahead. These are the types of effects global warming could have on our planet. Another important effect that global warming will bring is the loss and endangerment of many species. Did you know that 30 percent of all plant and animal species alive in the world today are at risk of extinction by the year 2050 if average temperatures rise more than 2 to 11.5 degrees Fahrenheit. These mass extinctions will be due to a loss of habitat through desertification, deforestation, and ocean warming. Many plants and animals will also be affected by the inability to adapt to our climate warming.

煤炭行业各项指标计划含义及其计算办法

煤炭生产统计有关指标计算办法摘编现将《煤炭工业计划统计常用指标计算办法》（1989年版）有关生产统计指标的相关规定和计算办法摘编，供生产统计人员学习参考。内容重点是原煤产量、掘进进尺、回采工作面利用、掘进工作面利用、采掘机械化程度、回采率等指标。一、原煤产量原煤指毛煤经过简单加工，拣除大块矸（大于50毫米）之后的煤炭。一切统计指标，都以原煤为对象。选前煤炭一般称毛煤。原煤产量必须加工拣选，实行选后计量，即拣出50毫米以上的矸石后，经验收合格后方可计算原煤产量。（一）原煤产量的计量 1、原煤计量形式原煤产量必须由矿井验收计量，不得按选后产品的数量倒算原煤产量。原煤计量方法由于提升运输方式不同而有不同手段。（1）矿井采用矿车运煤、提煤时，矿车计量以实际装载量计算。计算时扣除车底积煤；（2）箕斗和罐提煤的，以容积计算，定期（季）测定罐率和容积比重。全水分超过规定在容积比重中予以扣除（3）皮带提升的矿井安装电子（核子）称计量，定期测定比重和含矸率。（4）回采和掘进工作面煤炭计量采用盘方计量，即按照体积和原煤容重计算。回采产量=工作面采长×推进度×采高×原煤容重×工作面采出率掘进产量=煤巷（半煤岩）掘进毛断面×进尺×容重×掘进出煤系数原煤容重是本煤层实际测定的原煤比重（毛煤扣除含矸率后），与计算储量用的纯煤比重（视密度）不同。 2、月末核定产量的方法由于煤炭生产具有生产数量大且是连续性生产的特点，目前原煤计量手段都不同程度存在计算误差，必须在月末进行产量核定工作，保证原煤产量的准确性。一般采用“选前验收计量，月末核定产量”的方法。核定的方法：月末对原煤的实际库存量进行一次盘点，与通过本月逐日累计

煤炭分类及标准

煤炭分类及标准 Hessen was revised in January 2021

据统计，中国灰分小于10％的特低灰煤仅占探明储量的17％左右。大部分煤炭的灰分为10％～30％。硫分小于1％的特低硫煤占探明储量的43．5％以上，大于4％的高硫煤仅为2．28％。中国的炼焦用煤一般为中灰、中疏煤，低灰和低硫煤很少。炼焦用煤的灰分一般都在20％以上；硫分含量大于2％的炼焦用煤占20％以上。中国炼焦用煤的另一大特点是：硫分越高，煤的动结性往往越强，其可选性一般较差。中国褐煤多属老年褐煤。褐煤灰分一般为20％～30％。东北地区褐煤硫分多在1％以下，广东、广西、云南褐煤硫分相对较高，有的甚至高达8％以上。褐煤全水分一般可达20％～50％，分析基水分为10％~20％，低位发热量一般只有11．71～16．73MJ ／kg。中国烟煤的最大特点是低灰、低硫；原煤灰分大都低于15％，硫分小于1％。部分煤田，如神府、东胜煤田，原煤灰分仅为3％一5％，被誉为天然精煤。烟煤的第二个特点是煤岩组分中丝质组含量高，一般在40％以上，因此中国烟煤大多为优质动力煤。中国贫煤的灰分和硫分都较高，其灰分大多为15％-30％，流分在1．5％-5％之间。贫煤经洗选后，可作为很好的动力煤和气化用煤。中国典型的无烟煤和老年无烟煤较少，大多为三号年轻无烟煤，其主要特点是，灰分和硫分均较高，大多为中灰、中硫、中等发热量、高灰熔点，主要用作动力用煤，部分可作气化原料煤。

globalwarming教案

Teaching Plan Contents: Reading Book 6 Unit 4 Global warming I.Analysis of the Teaching Material This article is from a magazine about global warming, which illustrates how global warming has come about and different attitudes to its effects. The passage is long, abstract and far away from their life. What’s more, there are many mouthful professional terms, which increases students’ difficulty while reading, although they have some knowledge about global warming. II. Analysis of the Students Students from Senior Two are the students in an excellent level, who have good abilities to read and speak. This unit talks about global warming, which has been taught in Geography. It will help students understand the text better and I believe the students will be interested in this class. However,because they pay little attention to this topic in the daily life, they may have few desire to speak something about global warming. III. Teaching objectives 1. Knowledge objective 1) Enable the students to analyze how global warming has come about； 2) Get students know different attitudes towards global warming and its effects. 2. Competence objective Improve the students’ reading and speaking abilities. 3. Emotion objective 1) Develop student s’ teamwork. 2) Raise their awareness of global warming. IV. Important points Enable the students to understand how global warming has come about. V. D ifficult points Get the students understand how global warming has come about. Let the students understand the difficult sentences better. ①It is human activity that has caused this global warming rather than a random but natural phenomenon. (Line 6) ②All scientists subscribe to the view that the increase in the earth’s temperature is due t o the burning of fossil fuels like coal, natural gas and oil to produce energy. (Line 18) ③This is when small amounts of gases in the atmosphere, like carbon dioxide, methane and water vapour, trap heat from the sun and therefore warm the earth.(Line 26-29) VI. Teaching aids: Multimedia classroom, printed material VII. Teaching methods: Task-based teaching, communicative teaching method VIII.Teaching procedures: Step 1. Lead in and pre-reading (5 mins ) It’s reported that global temperatures continue to rise, making July 2016 the hottest month in the history of the earth. Did you feel extremely hot in July? When you felt hot, what did you do? Did you feel global temperatures going up quietly? Let’s look at a flash (global temperatures from 1850 to 2016). What information can you get? The earth is becoming warmer and warmer. Is it natural or caused by human being? Do you think what effects global warming will bring about? Is global warming beneficial or harmful? Today we’re going to read a magazine article about global warming. It will work out your puzzles. Please open your book and turn to P26. Today we are going to

煤炭各个指标之间的关系

煤炭各个指标之间的关系（神华煤炭化验设备）之前，我们了解到了：如何用神华煤炭化验设备去测量分析计算煤质各项指标的含量，那么这些煤炭质量指标之间又有什么关系呢煤的发热量、水分、灰分、挥发分、硫分、灰熔融性、G值、Y值之间有什么关系呢本文参考于：煤质检测分析新技术新方法与化验结果的审查计算实用手册，各项煤质指标间的相互关系，另外还有我神华煤炭化验设备公司专业技术人员提供的资料。 1.煤的工业分析各指标间的关系煤的工业分析项目，是了解和研究煤性质最基本指标，特别是水分、挥发分等指标，都能表征煤的不同煤化程度，之间均有显著的相关关系。此外，煤中矿物质的数量及其组分对煤的挥发分、发热量和真（视）相对密度等其他指标也都有显著影响。（2）原煤、精煤间的灰分关系：一般，洗选后的精煤灰分要比原煤的低，但灰分的降低幅度因煤的可选性而异。某些灰分不太高的年轻褐煤，往往用氯化锌重液洗后，其精煤的灰分反而比原煤的高。这是因为洗选过程中吸附造成的。（3）挥发分、焦渣特征和水分的关系：挥发分高低反映了煤的变质程度。焦渣特征在一定程度上反映了煤的粘结性和结焦性。 1）干燥无灰基挥发分和焦渣特征之间通常有下列关系：Vdaf≤l0%，焦渣特征为l～2号；Vdaf<13%，焦渣特征不超过4号；Vdaf>40%的褐煤，焦渣特征为1～2号；Vdaf=l8%～33%的炼焦用烟煤，焦渣特征为5～8号。 2）精煤干燥无灰基挥发分和原煤干燥无灰基挥发分之间，矿物质含量高的煤，其精煤干燥无灰

基挥发分往往稍小于原煤的。矿物质含量愈多，差值就愈大。但是，粘结性上，总是精煤高于原煤。 2.硫含量和工业分析指标间的关系一般，硫分高低和其它工业分析指标没有直接关系，但是，有机硫含量高的高硫煤，其发热量值常小于同一牌号的低硫煤。因为有机硫高的煤，其结构单元聚六碳环上的部分C、H被S取代，而C 和H的燃烧热值高。硫分和灰分间没有直接关系，但是，如果高硫煤中是以硫铁矿硫为主，则硫分高，其灰分产率也高；对于低硫煤，如果是有机硫为主，则情况相反。原煤和精煤中的硫含量变化有以下趋势：以无机硫为主的原煤，其精煤的硫含量低于原煤：以有机硫为主的原煤，其精煤的硫含量可能比原煤的还高。 3.胶质层最大厚度Y值与粘结指数G的关系烟煤的胶质层最大厚度Y值随粘结指数的增高而增高，但值在10~70之间时，Y值仅在4~15mm 之间变化，Y值为零的煤样，值比Y值灵敏得多。对值为95~105的煤，其Y值多在25~50mm之间，从而表明，在区分强粘结性煤时，Y值要比值灵敏得多。两者之间大致有如下关系：（1）Y值大于30mm的煤，其值均大于90；y值大于20mm的煤，其值一般均大于80；Y值小于15mm的煤，值一般小于80；Y值小于7mm的煤，值一般都在35以下。（2）值大于100的煤，其Y值一般都在25mm以上；值大于65的烟煤，Y值一般在10mm以上。（3）160多个煤样的计算结果表明，与Y值间的相关系数R值为，这表明两者呈显著的正比关系

煤炭分类及标准

中国煤炭分类国家标准表

判别煤炭质量优劣的指标很多，其中最主要的指标为煤的灰分含量和硫分含量。一般陆相沉积，煤的灰分、硫分普遍较低；海陆相交替沉积，煤的灰分、硫分普遍较高。中国煤炭灰分普遍较高，秦岭以北地区，晋北、陕北、宁夏、两淮、东北等地区，侏罗纪煤田为陆相沉积，煤的灰分一般为 10％～20％，有的在10％以下，硫分一般小于1％，东北地区硫分普遍小于0．5％。中国北方普遍分布的石灰纪、秦岭以南地区、湖南的黔阳煤系、湖北的梁山煤系等属海陆交替沉积的煤，灰分一般达15％～25％，硫分一般高达2％～5％。广西合山、四川上寺等地的晚二叠纪煤层属浅海相沉积煤，硫分可高达6％～10％以上。据统计，中国灰分小于10％的特低灰煤仅占探明储量的17％左右。大部分煤炭的灰分为10％～30％。硫分小于1％的特低硫煤占探明储量的43．5％以上，大于4％的高硫煤仅为2．28％。中国的炼焦用煤一般为中灰、中疏煤，低灰和低硫煤很少。炼焦用煤的灰分一般都在20％以上；硫分含量大于2％的炼焦用煤占20％以上。中国炼焦用煤的另一大特点是：硫分越高，煤的动结性往往越强，其可选性一般较差。中国褐煤多属老年褐煤。褐煤灰分一般为20％～30％。东北地区褐煤硫分多在1％以下，广东、广西、云南褐煤硫分相对较高，有的甚至高达8％以

上。褐煤全水分一般可达20％～50％，分析基水分为10％~20％，低位发热量一般只有11．71～16．73MJ／kg。中国烟煤的最大特点是低灰、低硫；原煤灰分大都低于15％，硫分小于1％。部分煤田，如神府、东胜煤田，原煤灰分仅为3％一5％，被誉为天然精煤。烟煤的第二个特点是煤岩组分中丝质组含量高，一般在40％以上，因此中国烟煤大多为优质动力煤。中国贫煤的灰分和硫分都较高，其灰分大多为15％-30％，流分在1．5％-5％之间。贫煤经洗选后，可作为很好的动力煤和气化用煤。中国典型的无烟煤和老年无烟煤较少，大多为三号年轻无烟煤，其主要特点是，灰分和硫分均较高，大多为中灰、中硫、中等发热量、高灰熔点，主要用作动力用煤，部分可作气化原料煤。

中国煤炭分类、煤质指标的分级

煤质指标的分级

中国煤炭分类 (2008-06-19 10:04:30) 中国煤炭分类：首先按煤的挥发分，将所有煤分为褐煤、烟煤和无烟煤；对于褐煤和无烟煤，再分别按其煤化程度和工业利用的特点分为2个和3个小类；烟煤部分按挥发分>10%~20%、>20%~28%、28%~37和>37%的四个阶段分为低、中、中高及高挥发分烟煤。关于烟煤粘结性，则按粘结指数G区分：0~5为不粘结和微粘结煤；>5~20为弱粘结煤；>20~50为中等偏弱粘结煤；>50~65为中等偏强粘结煤；>65则为强粘结煤。对于强粘结煤，又把其中胶质层最大厚度Y>25mm或奥亚膨胀度b>150%（对于Vdaf>28%的烟煤，b>220%）的煤分为特强粘结煤。在煤类的命名上，考虑到新旧分类的延续性，仍保留气煤、肥煤、焦煤、瘦煤、贫煤、弱粘煤、不粘煤和长焰煤8个煤类。在烟煤类中，对G>85的煤需再测定胶质层最大厚度Y值或奥亚膨胀度B值来区分肥煤、气肥煤与其它烟煤类的界限。当Y值大于25mm时，如Vdaf>37%，则划分为气肥煤。如Vdaf<37%，则划分为肥煤。如Y值<25mm，则按其Vdaf值的大小而划分为相应的其它煤类。如Vdaf>37%，则应划分为气煤类，如Vdaf>28%-37%，则应划分为1/3焦煤，如Vdaf在于28%以下，则应划分为焦煤类。这里需要指出的是，对G值大于100的煤来说，尤其是矿井或煤层若干样品的平均G 值在100以上时，则一般可不测Y值而确定为肥煤或气肥煤类。在我国的煤类分类国标中还规定，对G值大于85的烟煤，如果不测Y值，也可用奥亚膨胀度B值（%）来确定肥煤、气煤与其它煤类的界限，即对Vdaf<28%的煤，暂定b值>150%的为肥煤；对Vdaf>28%的煤，暂定b值>220%的为肥煤（当Vdaf值<37%时）或气肥煤（当Vdaf值>37%时）。当按b值划分的煤类与按Y值划分的煤类有矛盾时，则以Y值确定的煤类为准。因而在确定新分类的强粘结性煤的牌号时，可只测Y值而暂不测b值。（中国煤煤分类国家标准表）

2017_2018学年高中英语大题精做04Globalwarming含解析新人教版

Unit 4 Global warming I. 完形填空阅读下面短文，掌握其大意，然后从各小题所给的四个选项 (A、B、C和D) 中，选出最佳选项。 Today the scientific community is in almost total agreement that the earth’s climate is changing and that this represents a huge threat to the planet and to us. According to a survey, with only 69% accepting the earth is warming— only 1/4 Americans see global warming as a major threat, public opinion 1 the scientific conclusion. Climate scientists and campaigners have long debated how to better communicate the message to nonexperts so that climate science can be 2 into action. According to Christopher Rapley, the usual tactic(策略) of climate experts to provide the public wit h information isn’t 3 because ＂it does not address key potential causes.＂ We are all exposed to the evidence of climate change on an almost a daily basis. The information is almost 4 . Then what’s wrong? 5 our brains. Daniel Gilbert mentioned our brains’ failure to accu rately notice gradual change. Robert Gifford also __6__ the point about our brains’ difficulty in grasping climate change because of limited cognition and social __7__ with other people (＂Why should we change if X won’t?＂) .＂ A more powerful barrier is the 8 of perceived(感知的) be havioral control; ‘I’ m only one person; what can I do ?’ is certainly a big one.＂ For many, the first challenge will be in recognizing barriers 9 they can overcome them. But for those of us who understand that climate change is a problem yet make little effort to cut the number of overseas trips we make or the amount of meat we consume, neither the uncaring attitude nor denial really explains the 10 between our actions and beliefs. Lertzman has come to the conclusion that the conflict between __11 _ both the planet and our way of life is too painful to bear. ＂When we don’t 12 the pain of that, that’s when we get 13 and