Modeling and Optimization within Interacting Systems

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1 Modeling and Optimization within Interacting Systems Michael C. Ferris University of Wisconsin, Madison Optimization, Sparsity and Adaptive Data Analysis Chinese Academy of Sciences March 21, 2015 Ferris (Univ. Wisconsin) OSANDA 2015 Supported by DOE/USDA 1 / 38

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8 Idea and implementation Multiple agents interacting independently, along with shared resource Farmers (planting and management, leeching, CO2) Economy (supply, demand, money), Environment (bug index), Energy Use in schools, undergraduate classes and group of Ag/Econ experts Ferris (Univ. Wisconsin) OSANDA 2015 Supported by DOE/USDA 8 / 38

9 Idea and implementation Multiple agents interacting independently, along with shared resource Farmers (planting and management, leeching, CO2) Economy (supply, demand, money), Environment (bug index), Energy Use in schools, undergraduate classes and group of Ag/Econ experts Repeated game Single player not interesting - introduce bots Implement bots using GAMS Information in: same as a human player Key step: approximate other players actions/response function Different objectives Information out: planting and management decisions Point your google chrome browser at: fieldsoffuel.org Ferris (Univ. Wisconsin) OSANDA 2015 Supported by DOE/USDA 8 / 38

10 Aside: designing bots Bots receive same information as human players (see graphs and help) Only know own strategy Different objectives (economy, energy, environment, combination) Ferris (Univ. Wisconsin) OSANDA 2015 Supported by DOE/USDA 9 / 38

11 Aside: designing bots Bots receive same information as human players (see graphs and help) Only know own strategy Different objectives (economy, energy, environment, combination) Perennials: need history/look-ahead Runoff and bug index: need neighbors strategies Understand the economy/prices Prediction model for next 5 periods Solve multistage look-ahead MIP model (in real time) Distributed solution, each bot can use multiple cores Ferris (Univ. Wisconsin) OSANDA 2015 Supported by DOE/USDA 9 / 38

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13 Alternative: the big data model Collect states, and strategy decisions from real plays over time Use nearest neighbor to identify a small set of exemplars Randomly select an action from a selected exemplar to perform Perform an averaged action from exemplar set (worse performance) Test using cross validation and also deploy in real game Good CV performance, not used in real game at this time Can we use better schemes to exploit this accumulating data? Ferris (Univ. Wisconsin) OSANDA 2015 Supported by DOE/USDA 11 / 38

14 Alternative: the big data model Collect states, and strategy decisions from real plays over time Use nearest neighbor to identify a small set of exemplars Randomly select an action from a selected exemplar to perform Perform an averaged action from exemplar set (worse performance) Test using cross validation and also deploy in real game Good CV performance, not used in real game at this time Can we use better schemes to exploit this accumulating data? Data can be used to train a program to play like humans so that humans can reason about outcomes of multiple bot-played games Question: Can this be used to inform public policy decisions? Ferris (Univ. Wisconsin) OSANDA 2015 Supported by DOE/USDA 11 / 38

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16 (M)OPEC min x θ(x, p) s.t. g(x, p) 0 0 p h(x, p) 0 equilibrium min theta x g vi h p Solved concurrently Ferris (Univ. Wisconsin) OSANDA 2015 Supported by DOE/USDA 13 / 38

17 (M)OPEC min x θ(x, p) s.t. g(x, p) 0 0 p h(x, p) 0 x x θ(x, p) + λ T x g(x, p) 0 λ g(x, p) 0 0 p h(x, p) 0 equilibrium min theta x g vi h p Solved concurrently Requires global solutions of agents problems (or theory to guarantee KKT are equivalent) Theory of existence, uniqueness and stability based in variational analysis Ferris (Univ. Wisconsin) OSANDA 2015 Supported by DOE/USDA 13 / 38

18 MOPEC min θ i (x i, x i, p) s.t. g i (x i, x i, p) 0, i x i p solves VI(h(x, ), C) equilibrium min theta(1) x(1) g(1)... min theta(m) x(m) g(m) vi h p cons Reformulate optimization problem as first order conditions (complementarity) Use nonsmooth Newton methods to solve complementarity problem Solve overall problem Trade/Policy Model (MCP) using individual optimizations? Split model (18,000 vars) via region = + Gauss-Seidel, Jacobi, Asynchronous Ferris (Univ. Wisconsin) OSANDA 2015 Supported by DOE/USDA 14 / 38

19 General Equilibrium models (C) : max x k X k U k (x k ) s.t. p T x k i k (y, p) (P) : max p T g j (y j ) y j Y j (M) : max p 0 pt k x k k (I ) :i k (y, p) = p T ω k + j ω k j α kj p T g j (y j ) g j (y j ) s.t. p l = 1 l This is an example of a MOPEC Ferris (Univ. Wisconsin) OSANDA 2015 Supported by DOE/USDA 15 / 38

20 Special case: Nash Equilibrium Non-cooperative game: collection of players a A whose individual objectives depend not only on the selection of their own strategy x a C a = domθ a (, x a ) but also on the strategies selected by the other players x a = {x a : o A \ {a}}. Nash Equilibrium Point: x A = ( x a, a A) : a A : x a argmin xa C a θ a (x a, x a ). 1 for all a A, θ a (, x a ) is convex 2 C = a A C a and for all a A, C a is closed convex. Ferris (Univ. Wisconsin) OSANDA 2015 Supported by DOE/USDA 16 / 38

21 VI reformulation Define G : R N R N by G a (x A ) = a θ a (x a, x a ), a A where a denotes the subgradient with respect to x a. Generally, the mapping G is set-valued. Theorem Suppose the objectives satisfy (1) and (2), then every solution of the variational inequality x A C such that G(x A ) N C (x A ) is a Nash equilibrium point for the game. Moreover, if C is compact and G is continuous, then the variational inequality has at least one solution that is then also a Nash equilibrium point. Ferris (Univ. Wisconsin) OSANDA 2015 Supported by DOE/USDA 17 / 38

22 Strongly Convex (Generalized) Nash Equilibria min x 1 0 min x 2 0 No solution for θ 1: 1 2 x 1 2 θx 1 x 2 4x 1 s.t. x 1 + x x 2 2 x 1 x 2 3x 2 x 1 (x 2 ) = (θx 2 + 4) +, x 2 (x 1 ) = (x 1 + 3) + Solution 4 3 θ < 1: x 1 = 4+3θ 1 θ, x 2 = x Solution θ 4 3 : x 1 = 0, x 2 = 3 Jacobi works provided θ < 1, but diagonal dominance theory fails Ferris (Univ. Wisconsin) OSANDA 2015 Supported by DOE/USDA 18 / 38

23 Recast as a VI 1 1 θ x 1 4 M = 1 1 z = λ, q = x Mz + q + N C (z) 0 Mz + q z 0 Problem is not monotone (M not psd), so monotone operator splitting not possible New results (F/Rutherford/Wathen) show Jacobi/Gauss Seidel works based on Feingold/Varga (1962) M is an L-matrix, so Lemke method (PATH) solves the problem Ferris (Univ. Wisconsin) OSANDA 2015 Supported by DOE/USDA 19 / 38

24 Key point: models generated correctly solve quickly Here S is mesh spacing parameter S Var rows non-zero dense(%) Steps RT (m:s) : : : : 12 Convergence for S = 200 (with new basis extensions in PATH) Iteration Residual (+4) (+1) ( 2) ( 5) ( 11) Ferris (Univ. Wisconsin) OSANDA 2015 Supported by DOE/USDA 20 / 38

25 Extension to hierarchical models for policy analysis? The latest GTAP database represents global production and trade for 113 country/regions, 57 commodities and 5 primary factors. Data characterizes intermediate demand and bilateral trade in 2007, including tax rates on imports/exports and other indirect taxes. The core GTAP model is a static, multi-regional model which tracks the production and distribution of goods in the global economy. In GTAP the world is divided into regions (typically representing individual countries), and each region s final demand structure is composed of public and private expenditure across goods. Ferris (Univ. Wisconsin) OSANDA 2015 Supported by DOE/USDA 21 / 38

26 The Model The GTAP model (MOPEC) may be posed as a system of nonsmooth equations: in which: F + (w, z; t) = 0 w r is a vector of regional welfare levels z R N represents a vector( of endogenous ) economic variables, e.g. P prices and quantities, z =. Q t represents matrices of trade tax instruments import tariffs (tirs M) and export taxes (tirs X ) for each commodity i and region r Ferris (Univ. Wisconsin) OSANDA 2015 Supported by DOE/USDA 22 / 38

27 Optimal Sanctions Coalition member states strategically choose trade taxes which minimize Russian welfare: s.t. min t r :r C w rus F + (w, z; t) = 0 t r = t r r / C t M i,rus,r t M i,r,rus r C t X i,r,rus t X i,rus,r r C Ferris (Univ. Wisconsin) OSANDA 2015 Supported by DOE/USDA 23 / 38

28 Optimal Retaliation Russia choose trade taxes which maximize Russian welfare in response to the coalition actions: s.t. max w rus t rus F + (w, z; t) = 0 { ˆt t r = r r C t r r / C where ˆt r represents trade taxes for coalition countries (r C) from the optimal sanction calculation. Ferris (Univ. Wisconsin) OSANDA 2015 Supported by DOE/USDA 24 / 38

29 Coalition Member States for Illustrative Calculation usa United States anz Australia and New Zealand can Canada fra France deu Germany ita Italy jpn Japan gbr United Kingdom reu Rest of the European Union Ferris (Univ. Wisconsin) OSANDA 2015 Supported by DOE/USDA 25 / 38

30 Welfare Changes (% Hicksian EV) sanction retaliation tradewar rus C average can usa fra deu ita gbr reu anz jpn chn sau Ferris (Univ. Wisconsin) OSANDA 2015 Supported by DOE/USDA 26 / 38

31 Scenarios and Key Insights sanction If coalition states were to increases tariffs and export taxes on Russia to the same level which is currently applied by Russia on bilateral trade flows with the coalition, Russian welfare could be substantially impacted with no economic cost for any coalition members. retaliation Russia could respond to such sanctions by changing it s own trade taxes, but optimal retaliation largely results in a reduction rather than an increase in trade taxes on trade flows to and from coalition states. These tariff changes can only partially offset the adverse impact of the sanctions. tradewar If sanctions and retaliation were to result in an unconstrained trade war, Russia faces a drastic economic cost while the coalition countries could even be slight better off. Ferris (Univ. Wisconsin) OSANDA 2015 Supported by DOE/USDA 27 / 38

32 is Hydro-Thermal paper we will System illustrate (Philpott/F./Wets) the concepts using the voir and one thermal plant, as shown in Figure 1. Figure 1: Example hydro-thermal system. Ferris (Univ. Wisconsin) OSANDA 2015 Supported by DOE/USDA 28 / 38

33 Simple electricity system optimization problem SO: max d k,u i,v j,x i 0 s.t. W k (d k ) C j (v j ) + V i (x i ) k K j T i H U i (u i ) + v j d k, i H j T k K x i = xi 0 u i + hi 1, i H u i water release of hydro reservoir i H v j thermal generation of plant j T x i water level in reservoir i H prod fn U i (strictly concave) converts water release to energy C j (v j ) denote the cost of generation by thermal plant V i (x i ) future value of terminating with storage x (assumed separable) W k (d k ) utility of consumption d k Ferris (Univ. Wisconsin) OSANDA 2015 Supported by DOE/USDA 29 / 38

34 SO equivalent to CE Consumers k K solve CP(k): max d k 0 Thermal plants j T solve TP(j): max v j 0 Hydro plants i H solve HP(i): max u i,x i 0 W k (d k ) p T d k p T v j C j (v j ) p T U i (u i ) + V i (x i ) s.t. x i = xi 0 u i + hi 1 Perfectly competitive (Walrasian) equilibrium is a MOPEC CE: d k arg max CP(k), k K, v j arg max TP(j), j T, u i, x i arg max HP(i), i H, 0 p U i (u i ) + v j d k. i H j T k K Ferris (Univ. Wisconsin) OSANDA 2015 Supported by DOE/USDA 30 / 38

35 Agents have stochastic recourse? Two stage stochastic programming, x 1 is here-and-now decision, recourse decisions x 2 depend on realization of a random variable ρ is a risk measure (e.g. expectation, CVaR) SP: max c T x 1 + ρ[q T x 2 ] s.t. Ax 1 = b, x 1 0, T (ω)x 1 + W (ω)x 2 (ω) d(ω), x 2 (ω) 0, ω Ω. A T T W EMP/SP extensions to facilitate these models igure Constraints matrix structure of 15) problem by suitable subgradient methods in an outer loop. In the inner problem is solved for various righthand sides. Convexity of the master convexity of the value function in linear programming. In dual decom Ferris (Univ. Wisconsin) OSANDA 2015 Supported by DOE/USDA 31 / 38

36 Risk Measures VaR, CVaR, CVaR + and CVaR - Modern approach to modeling risk aversion uses concept of risk measures CVaR α : mean of upper tail beyond α-quantile (e.g. α = 0.95) Frequency VaR Probability 1 α CVaR Maximum loss Loss mean-risk, mean deviations from quantiles, VaR, CVaR Much more in mathematical economics and finance literature Optimization approaches still valid, different objectives, varying convex/non-convex difficulty Ferris (Univ. Wisconsin) OSANDA 2015 Supported by DOE/USDA 32 / 38

37 Two stage stochastic MOPEC CP(k): TP(j): HP(i): d 1 k v 1 j max max 0 0 W k ( d 1 k ) p 1 d 1 k p 1 v 1 j C j (v 1 j ) max p 1 U i (u 1 ui 1,x1 i 0 i ) s.t. x 1 i = x 0 i u 1 i + h 1 i, 0 p 1 i H U i ( u 1 i ) + j T v 1 j k K d 1 k Ferris (Univ. Wisconsin) OSANDA 2015 Supported by DOE/USDA 33 / 38

38 Two stage stochastic MOPEC CP(k): TP(j): HP(i): max d 1 k,d2 k (ω) 0 ( ) W k d 1 k p 1 dk 1 + ρ[w ( k d 2 k (ω) ) p 2 (ω)dk 2 (ω)] max p 1 v 1 j (ω) 0 j C j (vj 1 ) + ρ[p 2 (ω)vj 2 ( (ω) C j v 2 j (ω) ) ] v 1 j,v 2 max u 1 i,x1 i 0 u 2 i (ω),x2 i (ω) 0 p 1 U i (ui 1 ) + ρ[p 2 (ω)u i (ui 2 (ω)) + V i (xi 2 (ω))] s.t. x 1 i = x 0 i u 1 i + h 1 i, 0 p 1 i H x 2 i (ω) = x 1 i U i ( u 1 i ) + u 2 i (ω) + h 2 i (ω) j T v 1 j k K d 1 k Ferris (Univ. Wisconsin) OSANDA 2015 Supported by DOE/USDA 33 / 38

39 Two stage stochastic MOPEC CP(k): TP(j): HP(i): max d 1 k,d2 k (ω) 0 ( ) W k d 1 k p 1 dk 1 + ρ[w ( k d 2 k (ω) ) p 2 (ω)dk 2 (ω)] max p 1 v 1 j (ω) 0 j C j (vj 1 ) + ρ[p 2 (ω)vj 2 ( (ω) C j v 2 j (ω) ) ] v 1 j,v 2 max u 1 i,x1 i 0 u 2 i (ω),x2 i (ω) 0 p 1 U i (ui 1 ) + ρ[p 2 (ω)u i (ui 2 (ω)) + V i (xi 2 (ω))] s.t. x 1 i = x 0 i u 1 i + h 1 i, 0 p 1 i H 0 p 2 (ω) i H x 2 i (ω) = x 1 i U i ( u 1 i ) + u 2 i (ω) + h 2 i (ω) j T U i ( u 2 i (ω) ) + j T v 1 j k K d 1 k vj 2 (ω) dk 2 (ω), ω k K Ferris (Univ. Wisconsin) OSANDA 2015 Supported by DOE/USDA 33 / 38

40 Equilibrium or optimization? Each agent has its own risk measure Is there a system risk measure? Is there a system optimization problem? min i C(x 1 i ) + ρ i ( C(x 2 i (ω)) )???? Can we modify (complete) system to have a social optimum by trading risk? How do we design these instruments? How many are needed? What is cost of deficiency? Can we solve efficiently / distributively? Ferris (Univ. Wisconsin) OSANDA 2015 Supported by DOE/USDA 34 / 38

41 Example as MOPEC: agents solve a Stochastic Program Buy y i contracts in period 1, to deliver D(ω)y i in period 2, scenario ω Each agent i: min C(x 1 i ) + ρ i ( C(x 2 i (ω)) ) s.t. p 1 x 1 i + vy i p 1 e 1 i (budget time 1) p 2 (ω)x 2 i (ω) p 2 (ω)(d(ω)y i + e 2 i (ω)) (budget time 2) 0 v i y i 0 (contract) 0 p 1 i ( e 1 i xi 1 ) 0 (walras 1) 0 p 2 (ω) i ( D(ω)yi + e 2 i (ω) x 2 i (ω) ) 0 (walras 2) Ferris (Univ. Wisconsin) OSANDA 2015 Supported by DOE/USDA 35 / 38

42 Theory and Observations agent problems are multistage stochastic optimization models perfectly competitive partial equilibrium still corresponds to a social optimum when all agents are risk neutral and share common knowledge of the probability distribution governing future inflows situation complicated when agents are risk averse utilize stochastic process over scenario tree under mild conditions a social optimum corresponds to a competitive market equilibrium if agents have time-consistent dynamic coherent risk measures and there are enough traded market instruments (over tree) to hedge inflow uncertainty Otherwise, must solve the stochastic equilibrium problem Research challenge: develop reliable algorithms for large scale decomposition approaches to MOPEC Ferris (Univ. Wisconsin) OSANDA 2015 Supported by DOE/USDA 36 / 38

43 What is EMP? Annotates existing equations/variables/models for modeler to provide/define additional structure equilibrium vi (agents can solve min/max/vi) bilevel (reformulate as MPEC, or as SOCP) disjunction (or other constraint logic primitives) randvar dualvar (use multipliers from one agent as variables for another) extended nonlinear programs (library of plq functions) Currently available within GAMS Ferris (Univ. Wisconsin) OSANDA 2015 Supported by DOE/USDA 37 / 38

44 Conclusions MOPEC problems capture complex interactions between optimizing agents Policy implications addressable using MOPEC MOPEC available to use within the GAMS modeling system Stochastic MOPEC enables modeling dynamic decision processes under uncertainty Many new settings available for deployment; need for more theoretic and algorithmic enhancements Ferris (Univ. Wisconsin) OSANDA 2015 Supported by DOE/USDA 38 / 38

Modeling, equilibria, power and risk

Modeling, equilibria, power and risk Michael C. Ferris Joint work with Andy Philpott and Roger Wets University of Wisconsin, Madison Workshop on Stochastic Optimization and Equilibrium University of Southern