stochopt-long as part of stochastic optimization

Stochastic Optimization is
(almost) as easy as Deterministic
Optimization
Chaitanya Swamy
Joint work with David Shmoys
done while at Cornell University

Stochastic Optimization
• Way of modeling uncertainty.
• Exact data is unavailable or expensive – data is
uncertain, specified by a probability distribution.
Want to make the best decisions given this
uncertainty in the data.
• Applications in logistics, transportation models,
financial instruments, network design, production
planning, …
• Dates back to 1950’s and the work of Dantzig.

An Example
Components Assembly Products
Assemble To Order system
Demand is a random variable.
Want to decide inventory levels
now anticipating future demand.
Then get to know actual demand.
Can adjust inventory levels after
observing demand.
Excess demand  can buy more
stock paying more at the last minute
Low demand  can stock inventory
paying a holding cost

Two-Stage Recourse Model
Given : Probability distribution over inputs.
Stage I : Make some advance decisions – plan ahead
or hedge against uncertainty.
Observe the actual input scenario.
Stage II : Take recourse. Can augment earlier
solution paying a recourse cost.
Choose stage I decisions to minimize
(stage I cost) + (expected stage II recourse cost).

2-Stage Stochastic Facility
Location
Distribution over clients gives
the set of clients to serve.
client set D
facility
Stage I: Open some facilities in
advance; pay cost fi for facility i.
Stage I cost = ∑(i opened) fi .
stage I facility

Location
client set D
facility
stage I facility
How is the probability distribution on clients specified?
• A short (polynomial) list of possible scenarios
• Independent probabilities that each client exists
• A black box that can be sampled.

Location
facility
stage I facility
Actual scenario A = { clients to serve}, materializes.
Stage II: Can open more facilities to serve clients in A; pay
cost fi
A
to open facility i. Assign clients in A to facilities.
Stage II cost = ∑ fi
A
+ (cost of serving clients in A).
i opened in
scenario A

Want to decide which facilities to open in stage I.
Goal: Minimize Total Cost =
(stage I cost) + EA D [stage II cost for A].
Two extremes:
1. Ultra-conservative: plan for “everything” and buy only
in stage I.
2. Defer all decisions to stage II.
Both strategies may be sub-optimal.
Want to prove a worst-case guarantee.
Give an algorithm that “works well” on any instance,
and for any probability distribution.

Approximation Algorithm
Hard to solve the problem exactly – even special
cases are #P-hard.
Settle for approximate solutions.
Give polytime algorithm that always finds near-
optimal solutions.
A is a -approximation algorithm if,
•A runs in polynomial time.
•A(I) ≤ .OPT(I) on all instances I,
 is called the approximation ratio of A.

Previous Models Considered
• Dye, Stougie & Tomasgard:
– approx. algorithm for a resource provisioning problem.
– polynomial-scenario model.
• Ravi & Sinha (RS04):
– other problems in the polynomial-scenario model.
• Immorlica, Karger, Minkoff & Mirrokni:
– polynomial-scenario model and independent-activation model.
– proportional costs: (stage II cost) = (stage I cost),
e.g., fi
A
= .fi for each facility i, in each scenario A.
• Gupta, Pal, Ravi & Sinha (GPRS04):
– black box model: arbitrary probability distributions.
– Also need proportional costs.

Our Results
• Give the first approximation algorithms for
2-stage stochastic integer optimization problems
– black-box model
– no assumptions on costs.
For some problems improve upon previous results
obtained in more restricted models.
• Give a fully polynomial approximation scheme for a large
class of 2-stage stochastic linear programs (contrast with
Nesterov & Vial ‘00, Kleywegt, Shapiro & Homem De-Mello ‘01, Dyer,
Kanan & Stougie ‘02).
• Give a way to “reduce” stochastic optimization problems
to their deterministic versions.

Stochastic Set Cover (SSC)
Universe U = {e1, …, en }, subsets S1, S2, …, Sm
 U, set S has
weight wS.
Deterministic problem: Pick a minimum weight collection of
sets that covers each element.
Stochastic version: Set of elements to be covered is given by
a probability distribution.
– choose some sets initially paying wS for set S
– subset A  U to be covered is revealed
– can pick additional sets paying wS
A
for set S.
Minimize (w-cost of sets picked in stage I) +
EA U [wA
-cost of new sets picked for scenario A].

An Integer Program
For simplicity, consider wS
A
= WS for every scenario A.
pA : probability of scenario A  U.
xS : indicates if set S is picked in stage I.
yA,S : indicates if set S is picked in scenario A.
Minimize ∑S SxS + ∑AU pA ∑S WSyA,S
subject to,
∑S:eS xS + ∑S:eS yA,S ≥ 1 for each A  U, eA
xS, yA,S  {0,1} for each S, A.
A Linear Program
xS, yA,S ≥ 0 for each S, A
Exponential number of variables and exponential number
of constraints.

A Rounding Theorem
Assume LP can be solved in polynomial time.
Suppose for the deterministic problem, we have an
-approximation algorithm wrt. the LP relaxation, i.e.,
A such that A(I) ≤ .(optimal LP solution for I)
for every instance I.
e.g., “the greedy algorithm” for set cover is a
log n-approximation algorithm wrt. LP relaxation.
Theorem: Can use such an -approx. algorithm to get a
2-approximation algorithm for stochastic set cover.

Rounding the LP
Suppose we have an -approximation algorithm wrt. the LP
relaxation for the deterministic problem.
Let E = {e : ∑S:eS xS ≥ ½}.
So (2x) is a fractional set cover for the set E can round to get
an integer set cover S for E of cost ∑SS S ≤ (∑S 2SxS) .
S is the first stage decision.
Let (x,y) : optimal solution with cost OPT.
 for every element e, either
∑S:eS xS ≥ ½ OR in each scenario A : eA, ∑S:eS yA,S ≥ ½.

Sets
Elements
Rounding (contd.)
Set in S
Element in E
Consider any scenario A. Elements in A E are covered.
For every e AE, it must be that ∑S:eS yA,S ≥ ½.
So (2yA
) is a fractional set cover for AE  can round to
get a set cover of W-cost ≤ (∑S 2WSyA,S) .
A
Using this to augment S in scenario A, expected cost
≤ ∑SS S + 2.∑ AU pA (∑S WSyA,S) ≤ 2.OPT.

An -approx. algorithm for deterministic problem gives a
2-approximation guarantee for stochastic problem.
In the polynomial-scenario model, gives simple polytime
approximation algorithms for covering problems.
• 2log n-approximation for SSC.
• 4-approximation for stochastic vertex cover.
• 4-approximation for stochastic multicut on trees.
Ravi & Sinha gave a log n-approximation algorithm for
SSC, 2-approximation algorithm for stochastic vertex
cover in the polynomial-scenario model.
Rounding (contd.)

Let E = {e : ∑S:eS xS ≥ ½}.
So (2x) is a fractional set cover for the set E can round to get
an integer set cover S of cost ∑SS S ≤ (∑S 2SxS) .
S is the first stage decision.
Rounding the LP
Suppose we have an -approximation algorithm wrt. the LP
relaxation for the deterministic problem.
Let (x,y) : optimal solution with cost OPT.
 for every element e, either
∑S:eS xS ≥ ½ OR in each scenario A : eA, ∑S:eS yA,S ≥ ½.

A Compact Formulation
pA : probability of scenario A  U.
xS : indicates if set S is picked in stage I.
Minimize h(x) = ∑S SxS + f(x) s.t. xS ≥ 0 for each S
(SSC-P)
where, f(x) = ∑AU pAfA(x)
and fA(x) = min. ∑S WSyA,S
s.t. ∑S:eS yA,S ≥ 1 – ∑S:eS xS for each eA
yA,S ≥ 0 for each S.
Equivalent to earlier LP.
Each fA(x) is convex, so f(x) and h(x) are convex functions.

The General Strategy
1. Get a (1+)-optimal solution (x) to the convex
program using the ellipsoid method.
2. Convert fractional solution (x) to integer solution
– decouple stage I and stage II scenarios
– use -approx. algorithm for the deterministic problem
to solve subproblems.
Obtain a c.-approximation algorithm for the
stochastic integer problem.

If yi is infeasible, use violated inequality
to chop off infeasible half-ellipsoid.
The Ellipsoid Method
Ellipsoid  squashed sphere
Start with ball containing polytope P.
yi = center of current ellipsoid.
Min c.x subject to xP.
P

P
New ellipsoid = min. volume ellipsoid
containing “unchopped” half-ellipsoid.

If yi P, use objective function cut
c.x ≤ c.yi to chop off polytope, half-
ellipsoid.
c.x ≤ c.yi
P

ellipsoid.
P

P
x1, x2, …, xk: points lying in P. c.xk is a close to optimal value.
ellipsoid.
x1
x2
xk
x*

Ellipsoid for Convex Optimization
Min h(x) subject to xP.
P
If yi is infeasible, use violated inequality.
If yi P – how to make progress?
add inequality h(x) ≤ h(yi)? Separation
becomes difficult.
yi
h(x) ≤ h(yi)

Let d = subgradient at yi.
use subgradient cut d.(x–yi) ≤ 0.
Generate new min. volume ellipsoid.
P
d 
m
is a subgradient of h(.) at u, if for every v, h(v)-h(u) ≥ d.(v-u).
becomes difficult.
d
yi
h(x) ≤ h(yi)

P
d 
m
is a subgradient of h(.) at u, if for every v, h(v)-h(u) ≥ d.(v-u).
Let d = subgradient at yi.
use subgradient cut d.(x–yi) ≤ 0.
Generate new min. volume ellipsoid.
x1, x2, …, xk: points in P. Can show, mini=1…k h(xi) ≤ OPT+.
x*
x1
x2
becomes difficult.

Let d' = -subgradient at yi.
use -subgradient cut d'.(x–yi) ≤ 0.
P
x1, x2, …, xk: points in P. Can show, mini=1…k h(xi) ≤ OPT/(1-) + .
becomes difficult.
subgradient is difficult to compute.
d' 
m
is a -subgradient of h(.) at u, if vP, h(v)-h(u) ≥ d'.(v-u) – .h(u).
d'
yi
h(x) ≤ h(yi)

Subgradients and -subgradients
Vector d is a subgradient of h(.) at u,
if for every v, h(v) - h(u) ≥ d.(v-u).
Vector d' is an -subgradient of h(.) at u,
if for every vP, h(v) - h(u) ≥ d'.(v-u) – .h(u).
P = { x : 0 ≤ xS ≤ 1 for each set S }.
h(x) = ∑S SxS + ∑AU pA fA(x) = .x + ∑AU pA fA(x)
Lemma: Let d be a subgradient at u, and d' be a vector
such that dS – S ≤ d'S ≤ dS for each set S. Then,
d' is an -subgradient at point u.

Getting a “nice” subgradient
h(x) = .x + ∑AU pA fA(x)
fA(x) = min. ∑S WSyA,S
s.t. ∑S:eS yA,S ≥ 1 – ∑S:eS xS
eA
yA,S ≥ 0 S

h(x) = .x + ∑AU pA fA(x)
fA(x) = min. ∑S WSyA,S = max. ∑eA (1 – ∑S:eS xS) zA,e
s.t. ∑S:eS yA,S ≥ 1 – ∑S:eS xS s.t. ∑eA S zA,e ≤ WS
eA S
yA,S ≥ 0 S zA,e ≥ 0 eA

h(x) = .x + ∑AU pA fA(x)
fA(x) = min. ∑S WSyA,S = max. ∑e (1 – ∑S:eS xS) zA,e
s.t. ∑S:eS yA,S ≥ 1 – ∑S:eS xS s.t. ∑eS zA,e ≤ WS
eA S
yA,S ≥ 0 S zA,e = 0 eA, zA,e ≥ 0 e
Consider point u m
. Let zA  optimal dual solution for A at u. So
fA(u) = ∑e (1 – ∑S:eS uS) zA,e.
For any other point v, zA is a feasible dual solution for A. So
fA(v) ≥ ∑e (1 – ∑S:eS vS) zA,e.
Get that h(v) – h(u) ≥ ∑S (S – ∑AU pA ∑eS zA,e)(vS – uS) = d.(v-u) where
dS = S – ∑AU pA ∑eS zA,e. So d is a subgradient of h(.) at point u.

Computing an -Subgradient
Given point u 
m
. zA  optimal dual solution for A at u.
Vector d with dS = S – ∑AU pA ∑eS zA,e
= ∑AU pA(S – ∑eS zA,e) is a subgradient at u.
Goal: Get d' such that dS – S ≤ d'S ≤ dS for each S.
For each S, -WS ≤ dS ≤ S. Let  = maxS WS /S.
• Sample once from black box to get random scenario A.
Compute XS = S – ∑eS zA,e for each S.
E[XS] = dS and Var[XS] ≤ WS.
2
• Sample O(2
/2.log(m/)) times to compute d' such that
Pr[S, dS – S ≤ d'S ≤ dS] ≥ 1-
 d' is an -subgradient at u with probability ≥ 1-

Putting it all together
Min h(x) subject to xP.  Can compute -subgradients.
Run ellipsoid algorithm.
Given yi = center of current ellipsoid.
Continue with smaller ellipsoid.
If yi is infeasible, use violated
inequality as a cut.
If yi P use -subgradient cut.
P
x1
x2
xk
x*
Generate points x1, x2, …, xk in P. Return x = argmini=1…k h(xi).
Get that h(x) ≤ OPT/(1-) + .

Cannot evaluate h(.) – how to compute x = argmini=1…k h(xi)?
One last hurdle
xk
Will find point x in the convex hull of x1,…, xk such that h(x)
is close to mini=1…k h(xi).
x1
x2 x1 x2
Take points x1 and x2.
Do bisection search using -subgradients
to find point x on x1–x2 line segment with
value close to min(h(x1), h(x2)).
d' : -subgradient
d'.(x-y) ≥ 0
y

One last hurdle
xk
x2 x2
x1
x1

One last hurdle
xk
x2
x1
x2
Stop when search interval is small enough.
Set x = either end point of remaining segment.
x1
x x

Iterate using x and x3,…, xk updating x
along the way.
One last hurdle
xk
x2
x1
x

x
along the way.
One last hurdle
xk
x2
x1

One last hurdle
xk
x2
x1
along the way.

One last hurdle
xk
x2
x
x1
Can show that h(x) is close to mini=1…k h(xi).
along the way.

Finally,
Get solution x with h(x) close to OPT.
Sample initially to detect if OPT = Ω(1/) – this
allows one to get a (1+).OPT guarantee.
Theorem: (SSC-P) can be solved to within a factor
of (1 +) in polynomial time, with high probability.
Gives a (2log n+)-approximation algorithm for
the stochastic set cover problem.

A Solvable Class of Stochastic LPs
Minimize h(x) = w.x + ∑AU pAfA(x)
s.t. x m
, x ≥ 0, x P
where fA(x) = min. wA.yA + cA.rA
s.t. BA
rA ≥ jA
DA
rA + TA
yA ≥ lA
– TA
x
yA 
m
, rA 
n
, yA ≥ 0, rA ≥ 0.
Theorem: Can get a (1+)-optimal solution for this class of
stochastic programs in polynomial time.

Sample Average Approximation
Sample Average Approximation (SAA) method:
– Sample initially N times from scenario distribution
– Solve 2-stage problem estimating pA with frequency of occurrence
of scenario A
How large should N be?
Kleywegt, Shapiro & Homem De-Mello (KSH01):
– bound N by variance of a certain quantity – need not be
polynomially bounded even for our class of programs.
ShmoysS (recent work):
– show using -subgradients that for our class, N can be poly-
bounded.
Nemirovskii & Shapiro:
– show that for SSC with non-scenario dependent costs, KSH01 gives
polynomial bound on N for (preprocessing + SAA) algorithm.

Summary of Results
• Give the first FPRAS to solve a class of stochastic
linear programs.
• Obtain the first approximation algorithms for
2-stage discrete stochastic problems – no
assumptions on costs or probability distribution.
– 2log n-approx. for set cover
– 4-approx. for vertex cover (VC) and multicut on trees.
GPRS04 gave an 8-approx. for VC under
proportional costs in the black box model.

Results (contd.)
– 3.23-approx. for uncapacitated facility location (FL). Get
constant guarantees for several variants such as FL with
penalties, or soft capacities, or services.
Improve upon results of GPRS04, RS04 obtained in
restricted models.
– (1+)-approx. for multicommodity flow.
Immorlica et al. gave optimal algorithm for single-
commodity in polynomial “scenario” (demand) case.
• Give a general technique to lift deterministic
guarantees to stochastic setting.

Open Questions
• Practical Impact: Can one use -subgradients
in other deterministic optimization methods,
e.g., cutting plane methods? Interior-point
algorithms?
• Multi-stage stochastic optimization.
• Characterize which deterministic problems
are “easy to solve” in stochastic setting, and
which problems are “hard”.

stochopt-long as part of stochastic optimization

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