Multi-objective Optimization and
Stochastic Programming Models
We shall consider now various approaches to mathematical formulation of
stochastic programs.
Inspiration comes from multi-objective programming.
SP PROBLEM
Select the “best possible” decision which fulfills prescribed “hard”
constraints, say, x ∈ X where X ⊂ IRn is a closed nonempty set.
Outcome of a decision x is influenced by a random element ω of a general
nature whose realization is not known at the time of decision — scenario.
Random outcome of decision x is quantified by f (x, ω) and different
scenarios ω provide different optimal solutions,
x∗ (ω) ∈ arg minx∈X f (x, ω).
Scenario-based SP I.
ASSUME:
Finite set of considered scenarios, {ω s , s = 1, . . . , S}.
Methods of multi-objective programming suggest to choose a solution
efficient with respect to S objective functions f (x, ω s ), s = 1, . . . , S.
Such efficient solutions can be obtained, e.g., by minimization (or
maximization) of a weighted sum of f (x, ω s ), s = 1, . . . , S.
In our case, it is natural to use probabilities ps of scenarios ω s at the
place of weights ts and the problem to be solved is
min
x∈X
S
X
ps f (x, ω s ).
s=1
The result is the widely used expected value criterion.
Scenario-based SP II.
Notice that we get efficient solutions regardless the origin of probabilities
ps , e.g., for ps
the true probabilites,
subjective probabilities,
probabilities offered by experts,
equal probabilities obtained via simulation or coming from an
empirical probability distribution.
Similarly, using goal programming approach we may get the tracking
model, e.g.
S
X
min
ps |f (x, ω s ) − f (x∗ (ω s ), ω s )|.
x∈X
s=1
Models of SP with non-stochastic constraints
ASSUME:
Known, general probability distribution P of ω, EP denotes expectation.
Random elements appear only in the objective function f (x, ω),
set X is fixed.
Similarly as in financial applications, choice among various objective
functions which depend only on the probability distribution P,
NOT on observed scenarios.
In case of minimization, we have for example
minx∈X
minx∈X
minx∈X
minx∈X
minx∈X
f (x, ω̂), with ω̂ a point estimate of ω
EP f (x, ω), risk neutral expected value criterium
EP u(f (x, ω)), expected disutility criterium
EP f (x, ω) + λRP (f (x, ω) includes a risk measure RP
EP f (x, ω) + λvarP (f (x, ω), cf. Markowitz model
Random elements in constraints of (MP) I.
X (ω) = {x ∈ X0 : hj (x, ω) = 0 ∀j, gk (x, ω) ≤ 0 ∀k}
WE CANNOT WAIT WHICH SCENARIO OCCURS!
How to choose SET OF FEASIBLE DECISIONS INDEPENDENT OF
SCENARIOS?
AD HOC IDEA:
Replace ω by a fixed characteristics ω̂ which depends on P – point
estimate, mostly expectation EP ω. Solve deterministic problem with set
X (ω̂) = {x ∈ X0 : hj (x, ω̂) = 0 ∀j, gk (x, ω̂) ≤ 0 ∀k}
NOT THE BEST IDEA — example
min{x1 + x2 : αx1 + x2 ≥ 7, βx1 + x2 ≥ 4, x1 , x2 ≥ 0}
(α, β) random, uniformly distributed on [1, 4] × [1/3, 1]. Replace α, β by
their expected values 5/2 resp. 2/3 and solve the LP.
18
∗ ∗
Optimal solution x1∗ = 11
, x2∗ = 32
11 . Probability that (x1 , x2 ) satisfies the
random constraints is only 1/4! Not acceptable.
Random elements in constraints of (MP) II.
PERMANENTLY FEASIBLE or FAT DECISIONS
x ∈ X (ω) for all scenarios ω, or x ∈ X (ω) almost surely.
P{ω : x ∈ X (ω)} = 1.
The set X = ∩ω∈Ω X (ω) is small and it is often empty, e.g. for ω in
equations and P absolutely continuous. BUT — Robust Optimization.
RELAX THE REQUIREMENT −→ PROBABILITY or CHANCE
CONSTRAINTS
JOINT PROBABILITY CONSTRAINT
P(ω : x ∈ X (ω)) ≥ 1 − ,
with 0 ≤ ≤ 1 chosen by the decision maker.
Reliability type constraint; for absolutely continuous P can be used only
for random parameters in inequality constraits.
PROBLEM: convexity of the resulting set of feasible decisions. Can be
proved only for special structure of contraints and special probability
distribution, e.g. normal.
Individual Probability Constraints
Given probability treasholds 1 , . . . , m the feasible decisions are x ∈ X0
that fulfil m INDIVIDUAL PROBABILITY CONSTRAINTS
P(ω : gk (x, ω) ≤ 0) ≥ 1 − k , k = 1, . . . , m
(1)
Easy structure of problem, namely, if ω are right-hand sides of contraints,
i.e. gk (x, ω) = gk (x) − ωk .
Denote Fk marginal d.f. of ωk , uk (P) 100k % quantile of Fk −→ (1)
reformulated
Fk (gk (x)) ≤ k i.e. gk (x) ≤ uk (P)
For convex gk (x) — convex set of feasible decisions.
Notice: random rhs ωk replaced by a specified QUANTILE of marginal
pdf.
No more valid for joint probability constraints! Even for random rhs
special requirement on P needed, cf. log-concave or quasiconcave
probability distributions (Prékopa).
Joint Probability Constraints
THEOREM (Prékopa) Assume:
• gk (x, ω) ∀k are jointly convex in x, ω on IRn × IRl
• probability distribution P of ω is logarithmically concave on IRl .
THEN is the function
h(x) := P{ω : gk (x, ω) ≤ 0, ∀k}
logarithmically concave on IRn .
=⇒ convexity of the set X (P) := {x : h(x) ≥ α}
GENERALIZATIONS: to quasiconcave probability distributions −→
Applicable for a rich class of absolutely continuous probability
distributions
PROBLEMS: Joint convexity of gk (x, ω) −→
Theorem is applicable e.g. for gk (x, ω) = gk (x) − ωk , i.e. for
separable joint probabilistic constraints;
No direct application to discrete probability distributions
Random elements in constraints of (MP) — cont.
IDEA: Penalize discrepances and include the penalty into the objective
function, cf. Problem of Private Investor. Recall
Initial decision is independent of scenarios observed in future,
non-anticipativity.
It can be updated in dependence on observed scenarios later on to
fit the requirements as much as possible.
The cost of discrepace enters objective function; EXPECTED
PENALTY TERM.
Exist various models. EXAMPLE — NEWSBOY PROBLEM
Stochastic Linear Programs with Recourse I.
Linear program, some parameters random, e.g.
min{c> x : Ax = b, T(ω)x = h(ω)}
x≥0
Denote X0 := {x ≥ 0 : Ax = b}.
Penalize discrepance in T(ω)x = h(ω) when scenario ω occurs for a
chosen decision x ∈ X0 .
Simple way: For a fixed x ∈ X0 and for observed ω ∈ Ω, the discrepance
is
T(ω)x − h(ω) = y+ − y−
and is penalized by minimum possible cost
q+> y+ + q−> y−
subject to
T(ω)x − h(ω) = y+ − y− , y+ ≥ 0, y− ≥ 0.
(2)
SLP with Simple Recourse
This is SIMPLE RECOURSE, linear progam (2) is the 2nd stage
program.
Its value Q(x, ω) is finite and convex in x if q+ (ω) + q− (ω) ≥ 0 almost
surely.
The cost Q(x, ω) for discrepance, shortfall, etc. can be easily obtained as
q+> [T(ω)x − h(ω)]+ + q−> [T(ω)x − h(ω)]−
The full deterministic reformuation of the simple recourse problem is then
min{c> x + EP [q+> [T(ω)x − h(ω)]+ + q−> [T(ω)x − h(ω)]− }
subject to x ≥ 0 : Ax = b.
This was the problem of private investor and of newsboy and also the
first stochastic programs of Dantzig or Beale.
Stochastic Linear Programs with Recourse II.
General form of the 2nd stage problem:
minimize q(ω)> y
subject to
y ≥ 0, W(ω)y + T(ω)x = h(ω)
Its optimal value is Q(x, ω).
Generic form of the SLP with recourse is
minimize EP {c(ω)> x + Q(x, ω)}
subject to x ≥ 0 : Ax = b.
Recourse matrix W and recourse costs q are defined by the decision
maker.
(3)
Stochastic Linear Programs with Recourse III.
Special forms:
SLP with fixed recourse — W(ω) ≡ W, a fixed matrix;
SLP with complete recourse — W(ω) ≡ W, and the system Wy = z
has nonnegative solution y for arbitrary rhs. z;
SLP with simple recourse — only two types of discrepancies are
distinguished in each row of the system
T(ω)x = h(ω) −→ W = (I, −I) and I is the unit matrix.
SLP with relatively complete recourse — the 2nd stage problem has
almost surely a feasible solution for arbitrary x ∈ X0 , ω ∈ Ω.
Recourse costs can be random or fixed; the requirement — ∃ optimal
solution of the 2nd stage problem. Using LP duality, for SLP with fixed
complete recourse the condition is
{u : W> u ≤ q} =
6 ∅
Another item is existence of the expectation in (3)
Scenario-based SLP with recourse I.
Index by s = 1, . . . , S vectors and matrices q, W, T
possible realizations of the random coefficients, i.e.
and h of all
[qs , Ws , Ts , hs ] (componentwise)
and of corresponding second stage variables y,
assign probabilities ps to these realizations.
Stochastic program (3) is now
X
minimize
ps {c> x + Q(x, ωs )}
s
subject to x ≥ 0 : Ax = b
with Q(x, ωs ) the optimal value of the 2nd stage program
minimize q>
s y
subject to
y ≥ 0, Ws y + Ts x = hs
Scenario-based SLP with recourse II.
SP (3) can be rewritten as linear program of special dual block angular
structure
minimize c> x +
S
X
ps q>
s ys
(4)
s=1
subject to
Ax
T1 x
T2 x
..
.
=b
= h1
= h2
..
.
+ W1 y1
+W2 y2
..
.
TS x
+WS yS
x ≥ 0,
ys ≥ 0,
s = 1, . . . , S.
= hS
(5)
Comments
ys is the recourse action to take if scenario ωs occurs
It’s a BIG LINEAR program to solve
HOW BIG is it?
Immagine a problem: A Telecom company wants to expand its network
to meet unknown random demand. There are 86 unknown independent
demands, each may take one of 5 values.
−→ S = 586 = 4.77 × 1072 scenarios!
Scenario-based SLP with recourse III.
Alternative equivalent formulation of (4)– (5) is SCENARIO-SPLITTED
FORM
S
X
min
ps {c> x + q>
s ys }
s=1
subject to
Ax
Ts xs + Ws ys
x−
xs
x ≥ 0,
xs ≥ 0,
=b
= hs
= 0 ∀s
ys ≥ 0,
∀s
(6)
It helps to build procedures based on relaxation of the nonanticipativity
constraints
x = xs , s = 1, . . . , S.
(7)
SLP with recourse – Properties
Convenient property – CONVEXITY
Under primal and dual feasibility of the 2nd stage problem
minimize q(ω)> y
subject to
y ≥ 0, W(ω)y + T(ω)x = h(ω)
for fixed ω the optimal value Q(x, ω) is convex piecewise linear function
of x.
Convexity persists when taking expectation −→
we can take advantage of convex programs with linear constraints.
DIFFERENTIABILITY is not present for scenario-based programs, just
the piece-wise linear character persists.
Differentiability can be proved for absolutely continuous probability
distributions P in problems
min {c> x + Q(x)}
x∈X
with Q(x) := EP Q(x, ω).
Expected value of perfect information — EVPI
SP with recourse, see (3) is special instance of SP problems with
expected value criterium and fixed set of feasible decisions
min EP f (x, ω) for x ∈ X .
∗
HN
(8)
Let x (P) denote optimal solution of (8), v
optimal value.
Assume that all minima exist and compare optimal here-and-now value
v HN = EP f (x∗ (P), ω) of (8) with average of optimal values computed
for individual scenarios, i.e. with wait-and-see value
v WS := EP [min f (x, ω)].
x∈X
Evidently, for all x ∈ X , v WS = EP [minx∈X f (x, ω)] ≤ EP f (x, ω) =⇒
v WS ≤ min EP f (x, ω) = v HN
x∈X
and equality holds ⇐⇒ f (x∗ (P), ω) = minx∈X f (x, ω) almost surely.
Difference f (x∗ (P), ω) − minx∈X f (x, ω) is the value of perfect
information of using known realization ω and
EVPI := minx∈X EP f (x, ω) − EP [minx∈X f (x, ω)] = v HN − v WS
represents the expected value of perfect information.
Value of Stochastic Solution
Applications of SP are demanding.
WHAT DO WE GAIN IN COMPARISON WITH RESULTS OF
DETERMINISTIC PROLEMS OBTAINED BY REPLACING RANDOM
PARAMETERS ω BY THEIR EXPECTATIONS EP ω?
Denote EV= minx∈X f (x, EP ω) and x∗E an optimal solution;
EEV:= EP f (x∗E , ω) — expected result of using decision x∗E ;
HOW GOOD x∗E IS?
Value of stochastic solution,
VSS := EEV − v HN ≥ 0
There is no relationship between values of EVPI and VSS.
Application — Stochastic Dedicated Bond Portfolio
Management
Insert: Problem formulation and discussion
To solve the problem, one has to generate sensible SCENARIOS and
provide other model PARAMETERS.
∃ further generalizations to accommodate not only random (scenario
dependent) cash flows, but also liabilities and spread.
To include fully the trading possibilities one has to pass to the general
case of MULTISTAGE PROBLEMS:
Sequence of decisions is built along each of considered data trajectories
in such a way that decisions based on the same partial trajectory, on the
same history, are identical (nonanticipativity) and the outcome (e.g., the
expected gain or cost) of the decision process at the planning horizon is
the best possible.
OUTPUT ANALYSIS IS NECESSARY.