Risk-averse Stochastic Optimization: Models + Algorithms

1. Risk-averse Stochastic Optimization: Models + Algorithms Chaitanya Swamy University of Waterloo

2. Risk-averse Stochastic Optimization: Probabilistically-constrained models + Algorithms for Black-box Distributions Chaitanya Swamy University of Waterloo

3. 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).

4. Stage I: Open some facilities in advance; pay cost fifor 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 client set D

5. stage II facility Actual scenarioA = { clients to serve}, materializes. Stage II: Can open more facilities to serve clients in A; pay cost fiA to open facility i. Assign clients in A to facilities. Stage II cost = ∑ fiA + (cost of serving clients in A). i opened in scenario A 2-Stage Stochastic Facility Location Distribution over clients gives the set of clients to serve. Stage I: Open some facilities in advance; pay cost fifor facility i. Stage I cost = ∑(i opened) fi. facility stage I facility

6. Want to decide which facilities to open in stage I. Goal: Minimize Total Cost = (stage I cost)+ EA ÍD[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

7. Risk-averse stochastic optimization • E[.] measure does not adequately model the “risk” associated with stage-I decisions • Same E[.] value Þ 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

8. Modeling risk-aversion: attempt 1 Budget model Choose stage I decisions to minimize (stage I cost) + (expected stage II recourse cost) subject to (stage II cost of scenario A) ≤ B for every scenario A Gupta-Ravi-Sinha: considered stochastic Steiner tree in this budget model in the polynomial-scenario setting Budget model provides greatest degree of risk-aversion

9. Modeling risk-aversion: attempt 1 Budget model Choose stage I decisions to minimize (stage I cost) + (expected stage II recourse cost) subject to (stage II cost of scenario A) ≤ B for every scenario A Gupta-Ravi-Sinha: considered stochastic Steiner tree in this budget model in the polynomial-scenario setting • 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

10. Closely-related model Robust model Choose 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

11. Modeling risk-aversion: attempt 2 recall, budget model Choose stage I decisions to minimize (stage I cost) + (expected stage II recourse cost) 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

12. Modeling risk-aversion: attempt 2 Risk-averse budget model Choose stage I decisions to minimize (stage I cost) + (expected stage II recourse cost) subject to PrA[(stage II cost of scenario A) > B] ≤ r r: 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-r)-quantile of (stage II recourse cost)

13. 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 a-approximation algorithm if, • A runs in polynomial time. • A(I) ≤ a.OPT(I) on all instances I, • ais called the approximation ratio of A.

14. 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 Þcan 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)

15. 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 ……

16. Risk-averse Set Cover (RASC) Universe U = {e1, …, en }, subsets S1, S2, …, SmÍ 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 wS for set S • subset A Í U to be covered is revealed • can pick additional sets paying wSAfor set S. • Minimize(w-cost of sets picked in stage I)+ • EA ÍU [wA-cost of new sets picked for scenario A]. • subject to PrA ÍU [wA-cost for scenario A> B]≤ r

17. 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 Ato an extent of 1 Not 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]≤ r} is NOT a convex set. How to get an LP-relaxation?

18. An LP for fractional RASC For simplicity, consider wSA = WS for every scenario A. xS : indicates if set S is picked in stage I rA : 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 A Minimize ∑SwSxS +∑AÍU pA ∑S WS(yA,S + zA,S) subject to, ∑A pArA ≤r ∑S WS yA,S ≤ B for each A ∑S:eÎS xS + ∑S:eÎS yA,S + rA ≥ 1for each A, eÎA ∑S:eÎS xS + ∑S:eÎS (yA,S + zA,S)≥ 1for each A, eÎA xS, yA,S, zA,S ≥ 0 for each S, A.

19. Minimize ∑SwSxS +∑AÍU pA ∑S WS(yA,S + zA,S) subject to, ∑A pArA ≤r ∑S WS yA,S ≤ B for each A ∑S:eÎS xS + ∑S:eÎS yA,S + rA ≥ 1for each A, eÎA ∑S:eÎS xS + ∑S:eÎS (yA,S + zA,S)≥ 1for each A, eÎA xS, yA,S, zA,S ≥ 0 for each S, A. Coupling constraint • Exponential number of variables and exponential number of constraints. • The scenarios are no longerseparable: 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

20. Theorem 1: For any e,k>0, in time poly(1/ekr), can compute a first-stage soln.x that extends to an LP-soln. (x, {(yA,zA,rA)}A) of cost ≤ (1+e)OPT where ∑A pArA≤ r(1+k). Dependence on1/kris 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. w.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. • w.X + EA[W.YA]≤ 4aC, PrA[W.YA > 4aB] ≤ 2P • using any LP-based a-approx. algo. for DSC.

21. Rounding the LP Given 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:eÎS xS + ∑S:eÎS yA,S + rA ≥ 1for each A, eÎA ∑S:eÎS xS + ∑S:eÎS (yA,S + zA,S)≥ 1for each A, eÎA For every A, either we have rA ≥ 0.5 OR ∑S:eÎS xS + ∑S:eÎS yA,S ≥ 0.5 for each eÎA “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) Þw.x' + EA[fA(x')] ≤ 2C In scenario A, if rA ≤ 0.5, then (x', 2yA) covers A Þ fA(x') ≤ 2B So PrA[fA(x') > 2B] ≤ PrA[rA > 0.5] ≤ 2∑A pArA = 2P

22. Rounding (contd.) Rounding x' to an integer soln. to RASC: can use an a-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 ≤a.2C, and PrA[cost of A > a.2B] ≤ 2P • set cover, vertex cover, multicut on trees: Shmoys-S gave such a 2b-approx. algorithm using an LP-based b-approx. algo. for deterministic problemÞ 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

23. Solving the fractional-RASC LP: Sample Average Approximation • Sample 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? Wanted result: With poly-bounded N, x is an optimal solution to sample average problem Þ x is a near-optimal soln. to true problem with small blow-up of r

24. Solving the fractional-RASC LP Minimize ∑SwSxS +∑AÍU pA ∑S WS(yA,S + zA,S) subject to, ∑A pArA ≤r (*) ∑S WS yA,S ≤ B for each A ∑S:eÎS xS + ∑S:eÎS yA,S + rA ≥ 1for each A, eÎA ∑S:eÎS xS + ∑S:eÎS (yA,S + zA,S)≥ 1for each A, eÎA xS, yA,S, zA,S ≥ 0 for each S, A. 1) Lagrangify coupling constraint (*) to get a separable problem

25. Solving the fractional-RASC LP Minimize ∑SwSxS +∑AÍU pA ∑S WS(yA,S + zA,S) subject to, ∑A pArA ≤r (*) ] x D ≥ 0 ∑S WS yA,S ≤ B for each A ∑S:eÎS xS + ∑S:eÎS yA,S + rA ≥ 1for each A, eÎA ∑S:eÎS xS + ∑S:eÎS (yA,S + zA,S)≥ 1for each A, eÎA xS, yA,S, zA,S ≥ 0 for each S, A. 1) Lagrangify coupling constraint (*) to get a separable problem

26. Solving the fractional-RASC LP h(D;x) = w.x + ∑AÍU pA gA(D;x) MaxD ≥ 0[-Dr + min(∑SwSxS +∑AÍU pA (DrA+ ∑S WS(yA,S + zA,S)))] subject to, ∑S WS yA,S ≤ B for each A ∑S:eÎS xS + ∑S:eÎS yA,S + rA ≥ 1for each A, eÎA ∑S:eÎS xS + ∑S:eÎS (yA,S + zA,S)≥ 1for each A, eÎA xS, yA,S, zA,S ≥ 0 for each S, A. OPT(D) After Lagrangification, inner minimization problem becomes a separable 2-stage problem

27. Solving the fractional-RASC LP MaxD ≥ 0[-Dr + min(h(D;x) = w.x +∑AÍU pA gA(D;x))] 2) Argue that for each fixed D, can compute efficiently a “near-optimal” solution to inner-minimization problem 3) Use this to search for “right” value of the Lagrange-multiplier D:

28. Solving the fractional-RASC LP MaxD ≥ 0[-Dr + min(h(D;x) = w.x +∑AÍU pA gA(D;x))] 2) Argue that for each fixed D, can compute efficiently a “near-optimal” solution to inner-minimization problem 3) Use this to search for “right” value of the Lagrange-multiplier D: search is complicated by (i) only have approx. solns. for each D, (ii) cannot actually compute ∑A pArAbut 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 apply Crucial insight: For the search in 3) to work, suffices to prove the weak guarantee: can compute x s.t. h(D;x) ≈ (1+s)OPT(D)+ hD Weak enough that can show that sample-average-approximation works, by using approx.-subgradient proof technique (S-Shmoys)

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

30. 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.

31. Thank You.