Reformulation of chance constrained problems using penalty functions Martin Branda Charles University in Prague Faculty of Mathematics and Physics EURO XXIV July 11-14, 2010, Lisbon Martin Branda (MFF UK) Reformulation of CCP 2010 1 / 39
Contents 1 Reformulations of chance constrained problems 2 Asymptotic equivalence 3 Sample approximations using Monte-Carlo techniques 4 Numerical study and comparison Martin Branda (MFF UK) Reformulation of CCP 2010 2 / 39
Reformulations of chance constrained problems Contents 1 Reformulations of chance constrained problems 2 Asymptotic equivalence 3 Sample approximations using Monte-Carlo techniques 4 Numerical study and comparison Martin Branda (MFF UK) Reformulation of CCP 2010 3 / 39
Reformulations of chance constrained problems Optimization problem with uncertainty In general, we consider the following program with a random factor ω min {f (x) : x X, g i (x, ω) 0, i = 1,..., k}, (1) where g i, i = 0,..., k, are real functions on R n R n, X R n and ω R n is a realization of a n -dimensional random vector defined on the probability space (Ω, F, P). If P is known, we can use chance constraints to deal with the random constraints... Martin Branda (MFF UK) Reformulation of CCP 2010 4 / 39
Reformulations of chance constrained problems Multiple chance constrained problem ψ ɛ = min x X f (x), s.t. P ( g 11 (x, ω) 0,..., g 1k1 (x, ω) 0 ) 1 ε 1, P ( g m1 (x, ω) 0,..., g mkm (x, ω) 0 ). 1 ε m, (2) with optimal solution x ɛ, where we denoted ɛ = (ε 1,..., ε m ) with levels ε j (0, 1). The formulation covers the joint (k 1 > 1 and m = 1) as well as the individual (k j = 1 and m > 1) chance constrained problems as special cases. Martin Branda (MFF UK) Reformulation of CCP 2010 5 / 39
Reformulations of chance constrained problems Solving CCP In general, the feasible region is not convex even if the functions are convex, it is even not easy to check feasibility because it leads to computations of multivariate integrals. Hence, we will try to reformulate the chance constrained problem using penalty functions. Martin Branda (MFF UK) Reformulation of CCP 2010 6 / 39
Reformulations of chance constrained problems Consider the penalty functions ϑ j : R m R +, j = 1,..., m, continuous nondecreasing, equal to 0 on R m and positive otherwise, e.g. where u R k. ϑ 1,p (u) = k ( [ui ] +) p, p N i=1 ϑ 2 (u) = max [u i] +, 1 i k { = min t 0 : u i t 0, i = 1,..., k} Martin Branda (MFF UK) Reformulation of CCP 2010 7 / 39
Reformulations of chance constrained problems Penalty function problem Let p j denote the penalized constraints p j (x, ω) = ϑ j (g j1 (x, ω),..., g jkj (x, ω)), j Then the penalty function problem is formulated as follows [ ϕ N = min f (x) + N x X m j=1 ] E[p j (x, ω)] (3) with an optimal solution x N. In Y.M. Ermoliev, et al (2000) for ϑ 1,1 and m = 1. Martin Branda (MFF UK) Reformulation of CCP 2010 8 / 39
Reformulations of chance constrained problems Deterministic vs. stochastic penalty method Deterministic - penalizes the infeasibility with respect to the decision vector, cf. M.S. Bazara, et al. (2006). Stochastic - penalizes violations of the constraints jointly with respect to the decision vector and to the random parameter... Martin Branda (MFF UK) Reformulation of CCP 2010 9 / 39
Contents Asymptotic equivalence 1 Reformulations of chance constrained problems 2 Asymptotic equivalence 3 Sample approximations using Monte-Carlo techniques 4 Numerical study and comparison Martin Branda (MFF UK) Reformulation of CCP 2010 10 / 39
Asymptotic equivalence Assumptions (brief) Under the following assumptions, the asymptotic equivalence of the problems can be shown: Continuity of constraints and probabilistic functions. Compactness of the fixed set of feasible solutions. Existence of integrable majorants. Existence of a permanently feasible solution. No assumption on linearity or convexity! Martin Branda (MFF UK) Reformulation of CCP 2010 11 / 39
Assumptions Asymptotic equivalence Assume that X is compact, f (x) is a continuous function and (i) g ji (, ω), i = 1,..., k j, j = 1,..., m, are almost surely continuous; (ii) there exists a nonnegative random variable C(ω) with E[C 1+κ (ω)] < for some κ > 0, such that p j (x, ω) C(ω), j = 1,..., m, for all x X ; (iii) E[p j (x, ω)] = 0, j = 1,..., m, for some x X ; (iv) P(g ji (x, ω) = 0) = 0, i = 1,..., k j, j = 1,..., m, for all x X. Martin Branda (MFF UK) Reformulation of CCP 2010 12 / 39
Asymptotic equivalence Denote η = κ/(2(1 + κ)), and for arbitrary N > 0 and ɛ (0, 1) m put ε j (x) = P ( p j (x, ω) > 0 ), j = 1,..., m, m α N (x) = N E[p j (x, ω)], β ɛ (x) = ε η max j=1 m E[p j (x, ω)], j=1 where ε max denotes the maximum of the vector ɛ = (ε 1,..., ε m ) and [1/N 1/η ] = (1/N 1/η,..., 1/N 1/η ) is the vector of length m. THEN for any prescribed ɛ (0, 1) m there always exists N large enough so that minimization (3) generates optimal solutions x N which also satisfy the chance constraints (2) with the given ɛ. Martin Branda (MFF UK) Reformulation of CCP 2010 13 / 39
Asymptotic equivalence Moreover, bounds on the optimal value ψ ɛ of (2) based on the optimal value ϕ N of (3) and vice versa can be constructed: ϕ 1/ε η max (x N ) β ɛ(xn )(x ɛ(xn )) ψ ɛ(xn ) ϕ N α N (x N ), ψ ɛ(xn ) + α N (x N ) ϕ N ψ [1/N 1/η ] + β [1/N 1/η ] (x [1/N 1/η ] ), with lim α N(x N ) = N + lim ε j(x N ) = N + for any sequences of optimal solutions x N and x ɛ. lim β ɛ (x ɛ ) = 0 ε max 0 + Martin Branda (MFF UK) Reformulation of CCP 2010 14 / 39
Sample approximations using Monte-Carlo techniques Contents 1 Reformulations of chance constrained problems 2 Asymptotic equivalence 3 Sample approximations using Monte-Carlo techniques 4 Numerical study and comparison Martin Branda (MFF UK) Reformulation of CCP 2010 15 / 39
Sample approximations using Monte-Carlo techniques Let ω 1,..., ω S be an independent Monte Carlo sample of the random vector ω. Then, the sample version of the function q j is defined to be ˆq S j (x) = S 1 S ( I (0, ) pj (x, ω s ) ). (4) s=1 Martin Branda (MFF UK) Reformulation of CCP 2010 16 / 39
Sample approximations using Monte-Carlo techniques Finally, the sample version of the multiple jointly chance constrained problem (??) is defined as ˆψ γ S = min x X f (x), s.t. ˆq 1 S(x) γ 1,. ˆq m(x) S γ m, (5) where the levels γ j are allowed to be different from the original levels ε j. The sample approximation of the chance constrained problem can be reformulated as a large mixed-integer nonlinear program. Martin Branda (MFF UK) Reformulation of CCP 2010 17 / 39
Sample approximations using Monte-Carlo techniques Rates of convergence, sample sizes We will draw our attention to the case when the set of feasible solutions is finite, i.e. X <, which appears in the bounded integer programs, or infinite bounded. Using slight modification of the approach by S. Ahmed, J. Luedtke, A. Shapiro, et al. (2008, 2009), we obtain... Martin Branda (MFF UK) Reformulation of CCP 2010 18 / 39
Sample approximations using Monte-Carlo techniques Lower bound for the chance constrained problem We will assume that it holds γ j > ε j for all j. we can choose the sample size S to obtain that the feasible solution x is also feasible for the sample approximation with a probability at least 1 δ, i.e. S 2 min j {1,...,m} (γ j ε j ) 2 /ε j ln m δ, which corresponds to the result of S. Ahmed, et al (2008) for m = 1. Martin Branda (MFF UK) Reformulation of CCP 2010 19 / 39
Sample approximations using Monte-Carlo techniques Feasibility - finite X We will assume that it holds γ j < ε j for all j. Then it is possible to estimate the sample size S such that the feasible solutions of the sample approximated problems are feasible for the original problem, i.e. x X ɛ, with a high probability 1 δ S 1 2 min j {1,...,m} (γ j ε j ) 2 ln m X \ X ɛ. (6) δ If we set m = 1, we get the same inequality as J. Luedtke, et al (2008). Martin Branda (MFF UK) Reformulation of CCP 2010 20 / 39
Sample approximations using Monte-Carlo techniques Feasibility - bounded X We will assume that it holds γ j < ε j for all j and Lipschitz continuity of the penalized constraints, i.e. p j (x, ξ) p j (x, ξ) M j x x, x, x X, ξ Ξ, j, for some moduli M j > 0. Then it is possible to estimate the sample size S such that the feasible solutions of the sample approximated problems are feasible for the original problem, i.e. x X ɛ, with a high probability 1 δ S 2 min j {1,...,m} (ε j γ j ) 2 ( ln m δ + ln 2 min j {1,...,m} (ε j γ j ) + n ln ) 2Mmax D. τ If we set m = 1, we get the same inequality as J. Luedtke, et al (2008). Martin Branda (MFF UK) Reformulation of CCP 2010 21 / 39
Sample approximations using Monte-Carlo techniques 1. 2. 3. Stochastic Sample Solution programming approximation validation formulation (SA) Program with a random Chance constrained SA CCP Reliability factor problem (CCP) Penalty function SA PFP Reliability problem (PFP) Martin Branda (MFF UK) Reformulation of CCP 2010 22 / 39
Contents Numerical study and comparison 1 Reformulations of chance constrained problems 2 Asymptotic equivalence 3 Sample approximations using Monte-Carlo techniques 4 Numerical study and comparison Martin Branda (MFF UK) Reformulation of CCP 2010 23 / 39
Numerical study and comparison Numerical study and comparisons We will compare the ability of both sample approximated problems to generate a feasible solution of the original problem. Similar study was performed by J. Dupačová, et al. (1991). Martin Branda (MFF UK) Reformulation of CCP 2010 24 / 39
Numerical study and comparison Integer VaR and penalty function problems We consider 13 most liquid assets which are traded on the main market (SPAD) on Prague Stock Exchange. Weekly returns from the period 6th February 2009 to 10th February 2010 are used to estimate the means and the variance matrix. Suppose that the small investor trades assets on the mini-spad market. This market enables to trade mini-lots (standardized number of assets) with favoured transaction costs. Martin Branda (MFF UK) Reformulation of CCP 2010 25 / 39
Numerical study and comparison We denote Q i the quotation of the mini-lot of security i, f i the fixed transaction costs (not depending on the investment amount), c i the proportional transaction costs (depending on the investment amount), R i the random return of the security i, x i the number of mini-lots, y i binary variables which indicate, whether the security i is bought or not. Then, the random loss function depending on our decisions and the random returns has the following form n (R i c i )Q i x i + i=1 n f i y i. i=1 Martin Branda (MFF UK) Reformulation of CCP 2010 26 / 39
Numerical study and comparison Set of feasible solutions The set of feasible solutions contains a budget constraint and the restrictions on the minimal and the maximal number of mini-lots which can be bought, i.e. X = {(x, y) N n {0, 1} n B l n i=1 (1 + c i)q i x i + n i=1 f iy i B u, l i y i x i u i y i, i = 1,..., n}, where B l and B u are the lower and the upper bound on the capital available for the portfolio investment, l i > 0 and u i > 0 are the lower and the upper number of units for each security i. Martin Branda (MFF UK) Reformulation of CCP 2010 27 / 39
Numerical study and comparison Value at Risk problem The chance constrained portfolio problem can be formulated as follows min r (r,x,y) R X ( n P (R i c i )Q i x i + i=1 n i=1 ) f i y i r 1 ε, (7) this is in fact minimization of Value at Risk (VaR). Martin Branda (MFF UK) Reformulation of CCP 2010 28 / 39
Numerical study and comparison Penalty function portfolio problem Corresponding penalty function problem using the penalty ϑ 1,1 is [ min r + N E (r,x,y) R X n (R i c i )Q i x i + i=1 n + f i y i r]. (8) i=1 Setting N = 1/(1 ε) we obtain the CVaR problem, c.f. R.T. Rockafellar, S. Uryasev (2002). Martin Branda (MFF UK) Reformulation of CCP 2010 29 / 39
Numerical study and comparison Sample sizes - lower bound Table: Sample sizes - lower bound ε γ δ S 0.1 0.2 0.01 93 0.05 0.1 0.01 185 0.01 0.02 0.01 9211 0.1 0.2 0.001 139 0.05 0.1 0.001 277 0.01 0.02 0.001 13816 Martin Branda (MFF UK) Reformulation of CCP 2010 30 / 39
Numerical study and comparison Sample sizes - feasibility Bounded losses Table: Sample sizes - feasibility ε γ δ S 0.1 0.05 0.01 86496 0.05 0.025 0.01 348199 0.01 0.005 0.01 901792970 0.1 0.05 0.001 88338 0.05 0.025 0.001 355567 0.01 0.005 0.001 920213650 Martin Branda (MFF UK) Reformulation of CCP 2010 31 / 39
Numerical study and comparison Monte-Carlo simulation study Truncated normal distribution was used to model the random returns. 100 samples for each sample size S = 100, 250, 500, 750, 1000. Decreasing levels ε = 0.1, 0.05, 0.01. Increasing penalty parameters N = 1, 10, 100, 1000. Resulting mixed-integer linear programming problems were solved using GAMS and CPLEX... Reliability (feasibility) of the optimal solutions was verified on an independent sample of 10 000 realizations. Martin Branda (MFF UK) Reformulation of CCP 2010 32 / 39
Numerical study and comparison Reliability S γ min max mean st.dev 100 0.1 0.8844 0.9967 0.9592 0.0255 100 0.05 0.9054 0.9869 0.9516 0.0189 100 0.01 0.8939 0.9941 0.9456 0.0250 250 0.1 0.9546 0.9968 0.9824 0.0098 250 0.05 0.9545 0.9950 0.9820 0.0086 250 0.01 0.9555 0.9950 0.9807 0.0115 500 0.1 0.9744 0.9982 0.9903 0.0043 500 0.05 0.9744 0.9982 0.9903 0.0043 500 0.01 0.9726 0.9982 0.9906 0.0043 750 0.1 0.9849 0.9994 0.9952 0.0033 750 0.05 0.9849 0.9994 0.9952 0.0033 750 0.01 0.9866 0.9994 0.9953 0.0032 1000 0.1 0.9870 1.0000 0.9966 0.0025 1000 0.05 0.9870 1.0000 0.9966 0.0025 1000 0.01 0.9870 1.0000 0.9966 0.0025 Martin Branda (MFF UK) Reformulation of CCP 2010 33 / 39
Numerical study and comparison S N min max mean st.dev 100 1 0.7622 0.9480 0.8770 0.0303 100 10 0.8967 0.9976 0.9581 0.0220 100 100 0.8967 0.9976 0.9581 0.0219 100 1000 0.8967 0.9976 0.9581 0.0218 250 1 0.8330 0.9290 0.8888 0.0199 250 10 0.9495 0.9950 0.9788 0.0101 250 100 0.9571 0.9973 0.9841 0.0089 250 1000 0.9571 0.9973 0.9840 0.0089 500 1 0.8716 0.9270 0.9016 0.0134 500 10 0.9723 0.9955 0.9871 0.0044 500 100 0.9813 0.9996 0.9935 0.0033 500 1000 0.9813 0.9995 0.9934 0.0033 750 1 0.8697 0.9330 0.8990 0.0108 750 10 0.9785 0.9950 0.9878 0.0036 750 100 0.9890 0.9995 0.9957 0.0026 750 1000 0.9890 0.9993 0.9956 0.0026 1000 1 0.8739 0.9253 0.8976 0.0097 1000 10 0.9753 0.9964 0.9886 0.0038 1000 100 0.9900 0.9999 0.9966 0.0023 1000 1000 0.9900 0.9999 0.9966 0.0023 Martin Branda (MFF UK) Reformulation of CCP 2010 34 / 39
Conclusion Numerical study and comparison The penalty term decreases with increasing penalty parameter N and reduces violations of the stochastic constraint. The reliability of the obtained solutions increases with increasing levels γ and penalty parameters N for each sample size S. Both problems are also able to generate comparable solutions for the same sample sizes. Martin Branda (MFF UK) Reformulation of CCP 2010 35 / 39
References Numerical study and comparison S. Ahmed and A. Shapiro (2008). Solving chance-constrained stochastic programs via sampling and integer programming. In Tutorials in Operations Research, Z.-L. Chen and S. Raghavan (eds.), INFORMS. E. Angelelli, R. Mansini, M.G. Speranza (2008). A comparison of MAD and CVaR models with real features. Journal of Banking and Finance 32, 1188-1197. M.S. Bazara, H.D. Sherali, C.M. Shetty (2006). Nonlinear programming: theory and algorithms. Third Edition, John Wiley & Sons, New Jersey. Martin Branda (MFF UK) Reformulation of CCP 2010 36 / 39
References Numerical study and comparison M.B., J. Dupačová (2008). Approximations and contamination bounds for probabilistic programs. SPEPS 2008-13. M.B. (2010). Reformulation of general chance constrained problems using the penalty functions. SPEPS 2010-2. J. Dupačová, A. Gaivoronski, Z. Kos, T. Szantai (1991). Stochastic programming in water management: A case study and a comparison of solution techniques. European Journal of Operational Research 52, 28-44. Y.M. Ermoliev, T.Y. Ermolieva, G.J. Macdonald, and V.I. Norkin (2000). Stochastic optimization of insurance portfolios for managing exposure to catastrophic risks. Annals of Operations Research 99, pp. 207-225. Martin Branda (MFF UK) Reformulation of CCP 2010 37 / 39
References Numerical study and comparison J. Luedtke and S. Ahmed (2008). A sample approximation approach for optimization with probabilistic constraints. SIAM Journal on Optimization, vol.19, pp.674-699. R.T. Rockafellar, S. Uryasev (2002). Conditional Value-at-Risk for General Loss Distributions. Journal of Banking and Finance, 26, 1443-1471. A. Shapiro (2003). Monte Carlo Sampling Methods. In Stochastic Programming (A. Ruszczynski and A. Shapiro eds.), Handbook in Operations Research and Management Science Vol. 10, Elsevier, Amsterdam, 483-554. Martin Branda (MFF UK) Reformulation of CCP 2010 38 / 39
Numerical study and comparison Thank you for your attention. Martin Branda (MFF UK) Reformulation of CCP 2010 39 / 39