Risk-averse Stochastic Optimization: Models + Algorithms

Risk-averse Stochastic Optimization: Models +

Algorithms

Chaitanya SwamyUniversity of Waterloo

Risk-averse Stochastic Optimization: Probabilistically-

constrained models + Algorithms for Black-box

Distributions

Chaitanya SwamyUniversity of Waterloo

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

2-Stage Stochastic Facility Location

Distribution over clients gives the set of clients to serve.

facility

Stage I: Open some facilities in advance; pay cost fi for facility i.

Stage I cost = ∑(i opened) fi .

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 = ∑ fiA + (cost of serving

clients in A).

i opened inscenario A

stage II facility

Want to decide which facilities to open in stage I.

Goal: Minimize Total Cost = (stage I cost) + EA D [stage II cost for

A].How is the probability distribution

specified?

•A short (polynomial) list of possible scenarios

•Independent probabilities that each client exists

•A black box that can be sampled.

black-box setting

Risk-averse stochastic optimization

• E[.] measure does not adequately model the “risk” associated with stage-I decisions

• Same E[.] value same “risk involved”: given two solutions with same E[.] cost, prefer solution with more “assured” or “reliable” second-stage component (costs). E.g. portfolio investment

• Want to capture above notion of risk-averseness, where one seeks to avoid disaster scenarios

Modeling risk-aversion: attempt 1

Choose stage I decisions to minimize (stage I cost) + (expected stage II recourse cost)

Budget model provides greatest degree of risk-aversion

subject to (stage II cost of scenario A) ≤ B for every scenario AGupta-Ravi-Sinha: considered stochastic Steiner tree in this budget model in the polynomial-scenario setting

Budget model



Budget model provides greatest degree of risk-aversion, BUT

– limited modeling power: cannot get any approximation guarantees in black-box setting with bounded sample size

– overly conservative: protects every scenario regardless of its probability

subject to (stage II cost of scenario A) ≤ B for every scenario AGupta-Ravi-Sinha: considered stochastic Steiner tree in this budget model in the polynomial-scenario setting

Budget model

Closely-related modelChoose stage I decisions to minimize

(stage I cost) + (maximum stage II recourse cost)

• Dhamdhere et al. considered this model, again in the polynomial-scenario setting

• “Guessing” B = max. (stage II cost) “reduces” robust-problem to the budget problem

• Modeling issues: not clear how to even specify exponentially many scenarios

– Feige et al.: scenarios specified by cardinality constraint; seems rather stylized for stochastic optimization

– Will consider distribution-based robust model: scenario-collection = support of distribution

• Same drawbacks as in the budget model – no guarantees possible in black-box setting

Robust model



subject to (stage II cost of scenario A) ≤ B for every scenario A•For the budget-model, one can prove

approximation results if one is allowed to violate the budget-constraints with a small probability

•Can turn the above solution concept around and incorporate it into the model to arrive at the following new model

recall, budget model



subject to PrA[(stage II cost of scenario A) > B] ≤

: input – can tradeoff risk-averseness vs. conservatism

•Called probabilistically- or chance- constrained program

•Chance constraint called Value-at-Risk (VaR) constraint in finance literature: popular for risk-optimization in finance

•Related robust model: minimize (stage I cost) +

(1-)-quantile of (stage II recourse cost)

Risk-averse budget model

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.

Our Results•Obtain approximation algorithms for various risk-

averse budgeted (and robust) problems in the black-box setting: facility location, set cover, vertex cover, multicut on trees, min cut

•Give a fully polynomial approximation scheme for solving the LP-relaxations of a large class of risk-averse problems can use existing algorithms for deterministic or 2-stage version of problem to get approximation algorithm for risk-averse problem

•First approximation results for chance-constrained programs and black-box distributions (Kleinberg-Rabani-Tardos consider chance-constrained versions of bin-packing, knapsack but for specialized product distributions)

Related Work• Gupta et al.: gave a const.-approx. for stochastic

Steiner tree in the poly-scenario budget model

• Dhamdhere et al., Feige et al.: gave approx. algorithms for various problems in robust model with poly-scenarios, cardinality-collections

• So-Zhang-Ye: consider another risk measure called conditional VaR; give an approx. scheme for solving the LP-relaxations of problems in black-box setting– Can use our techniques to solve a generalization of

their model, where one has probabilistic budget constraints

• Lots of work in the standard 2-stage model: Dye et al., Ravi-Sinha, Immorlica et al., Gupta et al.+, Shmoys-S, S-Shmoys ……

Risk-averse Set Cover (RASC)

Universe U = {e1, …, en }, subsets S1, S2, …, Sm U, set S has weight wS.

Deterministic problem (DSC): Pick a minimum weight collection of sets that covers each element.Risk-averse budgeted version: Target set of elements to be covered is given by a probability distribution.

– choose some sets initially paying S for set S – subset A U to be covered is revealed – can pick additional sets paying wS

A for set S.

Minimize (-cost of sets picked in stage I) + EA U [wA-cost of new sets picked for scenario A].

subject to PrA U [wA-cost for scenario A > B] ≤

Fractional risk-averse set cover

Fractional risk-averse problem: can buy sets fractionally in stage I and in each scenario A to cover the elements in A to an extent of 1Not clear how to solve even the fractional problem in the polynomial-scenario setting.Why? The set of feasible solutions

{(x, {yA}A) : (x, yA) covers A for each scenario A,

PrA[∑S wAS yA,S > B] ≤ }

is NOT a convex set.How to get an LP-relaxation?

An LP for fractional RASCFor simplicity, consider wS

A = WS for every scenario A.xS : indicates if set S is picked in stage IrA : indicates if budget-constraint is NOT met for A{yA,S} : decisions in scenario A when budget-constraint is met for A{zA,S}: decisions in scenario A when budget-constraint is not met for AMinimize ∑S SxS + ∑AU pA ∑S WS(yA,S + zA,S)

subject to, ∑A pArA ≤

∑S WS yA,S ≤ B for each A

∑S:eS xS + ∑S:eS yA,S + rA ≥ 1 for each A, eA

∑S:eS xS + ∑S:eS (yA,S + zA,S) ≥ 1 for each A, eA

xS, yA,S, zA,S ≥ 0 for each S, A.

Minimize ∑S SxS + ∑AU pA ∑S WS(yA,S + zA,S)

subject to, ∑A pArA ≤




xS, yA,S, zA,S ≥ 0 for each S, A.

•Exponential number of variables and exponential number of constraints.

•The scenarios are no longer separable: i.e., a first-stage solution x alone is not enough to specify LP solution: need to specify the rAs – what does solving LP mean?– Contrast wrt. standard 2-stage model, or

fractional risk-averse problem

Coupling constraint

Theorem 1: For any ,>0, in time poly(1/), can compute a first-stage soln. x that extends to an LP-soln. (x, {(yA,zA,rA)}A) of cost ≤ (1+)OPT where ∑A pArA ≤ (1+).

Dependence on 1/ is unavoidable in black-box setting.

Theorem 2 (rounding theorem): Given a soln. x that extends to an LP-soln. (x, {(yA,zA,rA)}A) of cost C and ∑A pArA = P, can round x to

• a soln. x' for fractional RASC s.t. .x' + EA[opt. frac.

cost of A] ≤ 2C, PrA[opt. frac. cost of A > 2B] ≤ 2P[Can now use any LP-based “local” approx. for 2-stage SC

to round x'] •a soln (X, {YA}A) for (integer) RASC s.t.

.X + EA[W.YA] ≤ 4C, PrA[W.YA > 4B] ≤ 2P

using any LP-based -approx. algo. for DSC.

Rounding the LPGiven a soln. x that extends to an LP-soln. (x, {(yA,zA,rA)}A) of cost C and ∑A pArA = P

LP constraints: ∑S WS yA,S ≤ B for each A


∑S:eS xS + ∑S:eS (yA,S + zA,S) ≥ 1 for each A, eAFor every A, either we have

rA ≥ 0.5 OR ∑S:eS xS + ∑S:eS yA,S ≥ 0.5 for each eA

“Threshold rounding”: if rA ≥ 0.5, set r'A = 1, else r'A = 0; set x' = 2x

Let fA(x') = opt. fractional cost of scenario A given stage-I soln. x'

fA(x') ≤ W.(yA+zA) .x' + EA[fA(x')] ≤ 2C

In scenario A, if rA ≤ 0.5, then (x', 2yA) covers A fA(x')

≤ 2B

So PrA[fA(x') > 2B] ≤ PrA[rA > 0.5] ≤ 2 ∑A pArA = 2P

Rounding (contd.)Rounding x' to an integer soln. to RASC: can use an -approximation algorithm for 2-stage stochastic problem that is (i) LP-based, (ii) “local”, i.e., gives per-scenario cost guarantees, [(iii) can be implemented given only a first-stage solution]

to obtain integer solution (X, {YA}A) of cost ≤ .2C, and

PrA[cost of A > .2B] ≤ 2P

• set cover, vertex cover, multicut on trees: Shmoys-S gave such a 2-approx. algorithm using an LP-based -approx. algo. for deterministic problem get ratios of 4log n, 8, 8 respectively

• min s-t cut: can use O(log n)-approx. algorithm of Dhamdhere et al. for stochastic min s-t cut, which is local

• Also, facility location: not set cover, but very similar rounding; get 11-approx. using variant of Shmoys-S algorithm for 2-stage FL

Wanted result: With poly-bounded N,

x is an optimal solution to sample average problem x is a near-optimal soln. to true problem with small blow-up of

Solving the fractional-RASC LP: Sample Average

ApproximationSample Average Approximation (SAA) method:

– Sample some N times from distribution

– Estimate pA by qA = frequency of occurrence of scenario A = nA/N.

– Construct sample average LP, where pA is replaced by qA in LP

How large should N be?

Solving the fractional-RASC LP


subject to, ∑A pArA ≤ (*)




xS, yA,S, zA,S ≥ 0 for each S, A.1) Lagrangify coupling constraint (*) to get a separable problem



subject to, ∑A pArA ≤ (*) ] x ≥ 0∑S WS yA,S ≤ B for each A



xS, yA,S, zA,S ≥ 0 for each S, A.1) Lagrangify coupling constraint (*) to get a separable problem


Max≥ 0 [- + min ( ∑S SxS + ∑AU pA (rA + ∑S WS(yA,S +

zA,S)))]

subject to, ∑S WS yA,S ≤ B for each A



xS, yA,S, zA,S ≥ 0 for each S, A.After Lagrangification, inner minimization problem becomes a separable 2-stage problem

h(; x) = .x + ∑AU pA gA(; x)

OPT()


Max≥ 0 [- + min ( h(; x) = .x + ∑AU pA gA(; x))]2) Argue that for each fixed , can compute efficiently a “near-optimal” solution to inner-minimization problem

3) Use this to search for “right” value of the Lagrange-multiplier :


Max≥ 0 [- + min ( h(; x) = .x + ∑AU pA gA(; x))]2) Argue that for each fixed , can compute efficiently a “near-optimal” solution to inner-minimization problem

3) Use this to search for “right” value of the Lagrange-multiplier : search is complicated by (i) only have approx. solns. for each , (ii) cannot actually compute ∑A pArA but have to estimate it

Problems with 2): Cannot compute a “good” optimal solution; 2-stage problem does not fall into the solvable class in Shmoys-S, or Charikar-Chekuri-Pal – their arguments do not directly applyCrucial insight: For the search in 3) to work, suffices to prove the weak guarantee: can compute x s.t. h(; x) ≈ (1+)OPT() +

Weak enough that can show that sample-average-approximation works, by using approx.-subgradient proof technique (S-Shmoys)

Summary and Extensions• Although LP-relaxation of (fractional) problem is non-

separable, has exponential size, can still compute near-optimal LP-first-stage decisions: present an FPTAS– LP-first-stage decisions are sufficient to round and obtain

near-optimal solution to fractional problem, which can be further rounded using various known approx. algorithms.

– Many applications: set cover, vertex cover, facility location, min s-t cut, multicut on trees: obtain first approximation algorithms for chance-constraints + black-box model

• Get same results for (i) non-uniform budgets; (ii) risk-averse robust problems; (iii) simultaneous budget constraints, e.g., Pr[facility cost > BF or service cost > BS or total cost > B] ≤

• (iv) B=0 problem: interesting one-stage problem; choose initial decisions so as to satisfy “most” scenarios

Open Questions

•Approximation results for other problems in the risk-averse models.

•Models and algorithms for multi-stage risk-averse stochastic optimization (in black-box setting).

•Risk-averse stochastic scheduling.

•Other combinations of multiple probabilistic budget constraints.

Thank You.

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Risk-averse Stochastic Optimization: Models + Algorithms