MATHEMATICAL PRELIMINARIES Sample Clauses

MATHEMATICAL PRELIMINARIES. 2.5.3 Robust optimisation framework‌ There are different models to describe this uncertainty. In LP, the uncertainty model is (f , ai, bi) ∈ U, where U ∈ Rn is the uncertainty set which, for our purposes, will always be closed.. Assume f is a constant, the robust LP problem is shown as min f T x i subject to aT x ≤ bi, ∀(ai, bi ∈ U). (2.23) If f is also uncertain, the problem can be written as min t x,t subject to t ≥ f T x, i aT x ≤ bi, ∀(ai, bi, f ∈ U). (2.24) Robust conic program can be shown as min t x,t subject to t ≥ f T x, i bi − aT x ∈ K, ∀(ai, bi, f ∈ U). (2.25) Finally robust convex program [45] is formulated as min x f0(x, δ) (2.26) subject to fi(x, δ) ≤ 0, ∀δ ∈ U. The above robust convex program is convex if for any δ, fi(x, δ) is convex in x, for i = 1, 2, .., p. The objective and constraints must be satisfied for any occurrences of the data in the uncertainty set U. Since there are infinite number of realisations for this data, the uncertain optimisation problems are called semi-infinite problems. As it is known, these problems are hard to solve, and therefore, three different common approaches are considered to relax the semi-infinite problems in this area. In each approach, a known mechanism is exploited to reduce the number of constraints. These approaches are namely: • Stochastic model [46], [47], where the optimisation is to optimise the average (mathematical expectation) of the objective and constraints. In this case, the robust counterpart of the uncertain problem would be min Eδ[f0(x, δ)]

MATHEMATICAL PRELIMINARIES. Figure 2.5: Multi-antenna wireless communication system [7].

MATHEMATICAL PRELIMINARIES. The signal-to-interference-plus-noise ratio for any user i is expressed as SINR h w 2 | |i i = . (2.43) j=1,j/=i |h w |2 + σ2 A common class of optimal transmit downlink beamforming for multiple users is to find a set of wi that minimises the total transmit power while guaranteing all users’ SINR requirements γi, ∀i: min wi U Σ w wi H i (2.44) ΣU subject to |hiwi| j=1,j i j |h w |2 + σ2 ≥ γi, ∀1 ≤ i ≤ U. For simplicity, it is assumed that the set of γi in (2.44) is feasible. It can be verified that the SINR constraints in (2.44) are non-convex. In the next section, a technique to reformulate (2.44) in SOCP and SDP forms is presented.

MATHEMATICAL PRELIMINARIES. Using (2.20), the SOCP in (2.52) can be written in a SDP form as min W,p p γi  1 + 1 [HW]i,i eT HW σ  σ 1 + γi [HW]i,iI subject to  WHHHei 1  ≥ 0, ∀1 ≤ i ≤ U (2.53) p vecH (W) vec (W) pI ≥ 0. Solving (2.52) or (2.53) provides the optimal beamforming matrix W and the optimal downlink power as p2. Beamformer for user i can be obtained as the ith column of W.

MATHEMATICAL PRELIMINARIES. Section 2.1

MATHEMATICAL PRELIMINARIES. 2.1.1 Basic notation and probabilities LetZ,R,R +,Cdenote the set of integers, real numbers, non-negative real num- bers and complex numbers respectively. For a setS,|S|is the cardinality of that set.[1, n]is a shorthand for{i∈Z: 1≤i≤n}. We write := to assign $ the right-hand side to the left-hand side.a →Aindicates that an elementais ⟨A V ⟩ A →A selected uniformly at random from a setA, whereasa means that it is produced by the algorithm (probabilistically or not). In the case of interac- tive algorithms, we let , (x) denote the outcome of an interaction between