Problem Formulation

In this chapter we will consider the problem of calculating high-quality dual bounds for a more general class of MIP having the form

ζ : min

x,z tfpxq : Qx  z, x P X, z P Zu , (5.1)

where f is convex and continuously differentiable, QP Rqnis a block-diagonal matrix determining linear constraints Qx z, X € Rnis a closed and bounded set, and Z € Rqis a linear subspace. The vector xP X of decision variables is derived from the original decisions associated with a problem, while the vector z P Z of auxiliary variables are introduced to effect a decomposable structure in (5.1). In particular, the decomposable structure takes the form: 1) X ±m_i1Xi with Xi € Rni

closed and bounded and °m_i1ni  n; 2) fpxq 

°m

i1fipxiq where fi : R

ni ÞÑ R are convex and

differentiable for i  1, . . . , m; 3) Q has block diagonal structure with block diagonal components denoted as Qi P Rqini, i 1, . . . , m where

°m

i1qi  q, so that after setting z  pziqi1,...,m, where

for each i 1, . . . , m, zi P Rqi, we may write Qx z as Qixi  zi, i 1, . . . , m. This decomposable

structure is implicitly present throughout the chapter, although explicit referral to it is typically avoided where it is not needed.

Assumption 5.1 Problem (5.1) is feasible with finite optimal value.

The structure of (5.1) is sufficiently general to include two-stage SIP problems. In particular, (5.1) represents a two-stage SIP when the problem data of (5.1) is defined as follows:

• f is a linear function in x.

• qi  qj and ni  nj for all i and j in 1, . . . , m, and furthermore n ¡ q. qi is the number of

first-stage variables, while ni
qi is the number of second-stage variables.

• Construct Qi by horizontal concatenation of the identity matrix of size qiwith the zero matrix

of size qi pqi
niq. This links the first-stage variables one-to-one with variables in z.

• X is restricted by the linear constraints of the SIP on first- and second-stage variables. • Z  tpz1, . . . , zq1, zq1 1, . . . , zmq1q | zj  ziq1 j @i  1, . . . , m
1, @j P 1, . . . , q1u. The

definition of this linear subspace enforces the non-anticipativity constraint.

By a similar (albeit more complicated) construction process it can be shown that multi-stage SIP problems can be represented by (5.1) as well. Stochastic programs in which f is convex but non- linear, or in which X is a general compact (not necessarily convex) set, can also be represented by (5.1).

We develop a solution approach to solving the following relaxation of (5.1), ζCLD : min

x,z tfpxq : Qx  z, x P convpXq, z P Zu (5.2)

and its Lagrangian dual problem due to the relaxation of Qx z, ζCLD  sup

ω

φCpωq, (5.3)

which is based on the dual function φCpωq : min

x fpxq ω

J_{pQx
zq : x P convpXq, z P Z}(_{. (5.4)}

When f is linear, then φCpωq  φ pωq where φpωq : min

x,z fpxq ω

J_{pQx
zq, x P X, z P Z}(_{. (5.5)}

(That is, when f is linear, the role of X and convpXq are interchangeable.) Consequently, when f is linear, ζCLD  ζLD : sup_ωφpωq. However, when f is nonlinear, then in general, ζCLD ¤ ζLD.

89 Strict inequality is demonstrated with the following example. Let f : R2 _{ÞÑ R be defined by}

fpxq  px1
0.5q2 px2
0.5q2, X  t0, 1u  t0, 1u, and let Qx  z be defined to model the

constraints x1
z1  0 and x2
z2  0 where Z  tpz1, z2q : z1  z2u € R2. We see trivially

that ζCLD _{ 0, which is verified with the saddle point x}

1  x2  z1  z2  0.5 and ω  p0, 0q.

However, ζLD  0.5, which is verified with either of the saddle points x

1  x2  z1  z2  0 and

ω  p0, 0q, or x₁  x₂  z₁  z₂  1 and ω  p0, 0q. Thus, ζCLD   ζLD_.

Given that X is compact and f is continuous, in order for
8   φCpωq to hold, it is necessary and sufficient that the dual feasibility assumption

ωP ZK : υ P Rq : υJz  0 for all z P Z( (5.6) is maintained either by assumption or by construction. Under condition (5.6), the z term in defini- tion (5.4) vanishes, and we may compute

φCpωq  min

x fpxq ω

J_{Qx : xP convpXq}(_.

Consequently, φC becomes separable as

φCpωq  m ¸ i1 φC_i pωiq, where φC i pωiq : minx fipxiq ωiJQixi : xi P convpXiq ( and ω  pω1, . . . , ωmq P Rq1     Rqm has

a block structure compatible with the block diagonal structure of Q.

Given that X is closed and bounded (thus so is convpXq), and (5.2) is assumed to be feasible, then in order to guarantee that the maximum in (5.3) is realised for some ω P ZK, we assume a constraint qualification such as Slater’s condition. In other words, we assume that there exists px_{, z}_{q such that x} _{P intpconvpXqq and Qx} _{ z}_{, where intpq returns the topological interior of}

the set argument. If convpXq is polyhedral, then even this Slater’s condition is not required.