Nonlinear chance-constrained problems with applications to hydro - - PowerPoint PPT Presentation

Nonlinear chance-constrained problems with applications to hydro scheduling

Enrico Malaguti

DEI - University of Bologna Joint work with Andrea Lodi◦, Giacomo Nannicini∗, Dimitri Thomopulos◦

• DEI - University of Bologna

∗Singapore University of Technology and Design

Aussois, January 7th, 2015

(slides stolen from Giacomo)

Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 1 / 29

1

The mid-term hydro scheduling problem

2

Problem decomposition and branch-and-cut

3

A separation routine

4

Finiteness

5

Computational experiments

Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 2 / 29

Scheduling for a hydro network

Motivating problem: Hydro power generation requires scheduling the operation of multiple power plants in advance. The decision maker must determine how to produce electrical power to maximize profit taking into account the demand. All decisions affect the future state of the network, because water released from upstream reservoirs flows into downstream reservoirs.

Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 3 / 29

Scheduling for a hydro network

Motivating problem: Hydro power generation requires scheduling the operation of multiple power plants in advance. The decision maker must determine how to produce electrical power to maximize profit taking into account the demand. All decisions affect the future state of the network, because water released from upstream reservoirs flows into downstream reservoirs. Uncertainty is a complicating factor: Rainfall cannot be accurately predicted beyond a few days. Aggregated demand can be partially inferred from historical data, but exogenous factors (e.g. weather) make it uncertain. Energy prices fluctuate over time.

Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 3 / 29

The uncertainty model

What we know: [Baslis and Bakirtzis, 2011] assume that forecasts for aggregated demand and precipitations are available as random parameters with discrete support and discrete probabilities. To optimize over a long enough time period (i.e. six months), they assume that random parameters in some time period depend on the realization in the previous period. This gives rise to a scenario tree, where a scenario is a realization of the random parameters.

Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 4 / 29

The scenario tree

demand: low inflow: very low

5 inflows, 3 demands

Stage 1: month 1

3 inflows, 2 demands

Stage 2: months 2−4 Stage 3: months 5−7

In total, there are 90 scenarios.

Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 5 / 29

Main characteristics of the problem

Problem structure: Multi-stage: at each stage we must decide how much water to release from the reservoirs (determines how much energy we can produce). Underlying flow network structure: water flows between reservoirs. The constraints impose lower and upper bounds on water levels, maximum water released, and minimum production requirements.

Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 6 / 29

Main characteristics of the problem

Problem structure: Multi-stage: at each stage we must decide how much water to release from the reservoirs (determines how much energy we can produce). Underlying flow network structure: water flows between reservoirs. The constraints impose lower and upper bounds on water levels, maximum water released, and minimum production requirements. The difficult part: The production function is nonlinear: at a first approximation concave, but in fact nonconvex. The inflows and the demands are random parameters: the profit over the entire time horizon is therefore nondeterministic. Prices are given by a step function depending on the amount produced and the energy market: modeling this requires discrete variables.

Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 6 / 29

Dealing with uncertainty

The decision-making model: As time moves forward, new information comes in, and the decision-maker may re-evaluate her options. At the time of solving the problem, we should be concerned with current decisions: future decisions can be determined at a later stage. Since profit is uncertain, what do we optimize?

Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 7 / 29

Dealing with uncertainty

The decision-making model: As time moves forward, new information comes in, and the decision-maker may re-evaluate her options. At the time of solving the problem, we should be concerned with current decisions: future decisions can be determined at a later stage. Since profit is uncertain, what do we optimize? Why not worst case: We could optimize a lower bound on the profit in the worst-case scenario. This is typically a very conservative bound. We would like a more flexible model where the trade-off between robustness and profit can be controlled.

Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 7 / 29

Optimizing quantiles

Interpreting the profit: Assume there are k possible future scenarios ξ1, . . . , ξk. Let ϕ(x = ¯ x, ξ = ξi) be the profit obtained in the i-th scenario with decision variables ¯ x. If ¯ x is given, we can compute the distribution of ϕ. Our formulation: choose x in order to maximize the α-quantile of ϕ. The parameter α controls the level of risk.

Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 8 / 29

1

The mid-term hydro scheduling problem

2

Problem decomposition and branch-and-cut

3

A separation routine

4

Finiteness

5

Computational experiments

Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 9 / 29

Formulation as chance-constrained problem

Feasible region and scenarios: Some constraints of the problem are deterministic: scenario-independent, x ∈ X. Other constraints are nondeterministic: scenario-dependent. Let C(ξ) be the feasible region of the problem given scenario ξ. We want to take decisions in such a way that the probability associated with satisfied scenarios is large. max{lower bound on profit : Pr(x ∈ C(ξ)) ≥ 1 − α, x ∈ X}

Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 10 / 29

Basic idea for the formulation

Linear case (for simplicity): Variables: general-variables x, scenario-variables y. Introduce binary variables z to determine which scenarios are satisfied. A possible but impractical formulation is: min cx s.t.: Ax ≥ b A1x + H1y1 ≥ b1 − Mz1 A2x + H2y2 ≥ b2 − Mz2 . . . . . . Akx + HkyK ≥ bk − Mzk p1z1 + p2z2 + . . . + pkzk ≤ α x, y1, y2, . . . yk ≥ z1, z2, . . . zk ∈ {0, 1}.

Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 11 / 29

Decomposing the problem

Master problem: First and second stage decisions: x (because of nonanticipativity). A variable xϕ to indicate the α-quantile of the profit, which is maximized. Constraints that are independent of the realization of the random parameters: x ∈ X. Scenario subproblems, one for each scenario: First and second stage decisions x, third stage decisions: y. A constraint: xϕ ≤ profit for current scenario. Scenario-dependent constraints. Let Cx,y(ξi) be the feasible region of a scenario, and define Cx(ξi) := ProjxCx,y(ξi). So ˆ x is feasible for scenario i if ˆ x ∈ Cx(ξi). Basic idea: Generate solutions for the master. If they are not feasible for enough scenarios to satisfy the joint chance-constraint, cut them off.

Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 12 / 29

Benders decomposition and Branch-and-Cut

An exact solution approach: A Branch-and-Cut was proposed by [Luedtke, 2013] for this type of problems with finite support – some conditions must be satisfied. It uses a separation routine for the scenario subproblems, combined with the variables zi. We can enforce the constraint zi = 0 ⇒ x ∈ Cx(ξi) using an approach similar to [Codato and Fischetti, 2006].

Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 13 / 29

A Branch-and-Cut approach: overview

Assumptions: The scenario subproblems are polyhedra with the same recession cone. The objective function of the master problem is “well-behaved”, e.g. linear or convex over a compact.

Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 14 / 29

A Branch-and-Cut approach: overview

Assumptions: The scenario subproblems are polyhedra with the same recession cone. The objective function of the master problem is “well-behaved”, e.g. linear or convex over a compact. The algorithm in a nutshell: Add the zi variables to the master problem and perform Branch-and-Bound branching on the zi’s. At each node: if zi = 0 but the solution ˆ x does not belong to Cx(ξi), separate ˆ x from Cx(ξi) via an inequality αx ≥ β. Compute the rhs ¯ βj that makes αx ≥ ¯ βj valid for every scenario j = 1, . . . , k. This is just an LP under our assumptions. Add to the master problem the cut αx ≥ ρ(¯ β, z), where ρ(¯ β, z) is a linear function that adjusts the rhs to make the cut valid for the scenarios for which zi = 0.

Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 14 / 29

Separation

For linear programs: [Luedtke, 2013] discusses the case where the master and the scenario subproblems are linear programs (before the addition of the zi’s). In this case a cut separating ˆ x from Cx(ξ) can be obtained solving a linear program by using Benders’s approach.

Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 15 / 29

Separation

For linear programs: [Luedtke, 2013] discusses the case where the master and the scenario subproblems are linear programs (before the addition of the zi’s). In this case a cut separating ˆ x from Cx(ξ) can be obtained solving a linear program by using Benders’s approach. What if Cx,y(ξ) is a nonlinear problem? And integer?

Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 15 / 29

1

The mid-term hydro scheduling problem

2

Problem decomposition and branch-and-cut

3

A separation routine

4

Finiteness

5

Computational experiments

Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 16 / 29

Some notation for the nonlinear case

Preliminaries: We want to solve: max{cx : Pr(x ∈ Cx(ξ)) ≥ 1 − α, x ∈ X}. Remember that Cx(ξ) = ProjxCx,y(ξ), where: Cx,y(ξ) := {(x, y) : gi(x, y) ≤ 0, i = 1, . . . , m}, and the gi’s are convex. Given a solution ˆ x for the master problem, we must answer the question: does there exist ˆ y such that (ˆ x, ˆ y) ∈ Cx,y(ξ)? If not, find a separating constraint to cut ˆ x from the master.

Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 17 / 29

Some notation for the nonlinear case

Preliminaries: We want to solve: max{cx : Pr(x ∈ Cx(ξ)) ≥ 1 − α, x ∈ X}. Remember that Cx(ξ) = ProjxCx,y(ξ), where: Cx,y(ξ) := {(x, y) : gi(x, y) ≤ 0, i = 1, . . . , m}, and the gi’s are convex. Given a solution ˆ x for the master problem, we must answer the question: does there exist ˆ y such that (ˆ x, ˆ y) ∈ Cx,y(ξ)? If not, find a separating constraint to cut ˆ x from the master. Generalized Benders decomposition: [Geoffrion, 1972] provides an

• approach. However:

Strong assumptions on the separability of the gi’s in x and y; The cut is in general nonlinear.

Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 17 / 29

Separating hyperplanes

Beyond Geoffrion’s approach: We would like to find a linear cut in the x space (the y variables are scenario-dependent). Exploiting convexity: We can take advantage of the fact that Cx,y(ξ) is assumed convex. Hence, Cx(ξ) is convex. If Cx,y(ξ) is not feasible for a given ˆ x, it means that ˆ x ∈ Cx(ξ). We want to find a maximally violated hyperplane separating ˆ x from Cx(ξ).

Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 18 / 29

Separating hyperplanes

x2 Cx,y(ξ) ˆ x ProjxCx,y(ξ) y x1 arg min

(x,y)∈Cx,y(ξ) (ˆ

x, 0) − (x, y)x

Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 19 / 29

A separation routine

Assumptions: Define the problem: min

(x,y)∈Cx,y(ξ)

1 2x − ˆ x2

x

(P) Assume that (P) satisfies constraint qualification at all its optima.

Theorem: KKT-based separation

Assume ˆ x ∈ Cx(ξ). Let (¯ x, ¯ y) be the optimal solution to (P), and µi, i = 1, . . . , m a set of the corresponding optimal KKT multipliers. Let I := {i ∈ {1, . . . , m} : gi(¯ x, ¯ y) = 0}. Then, the hyperplane:

• i∈I

µi∇gi(¯ x, ¯ y)

• ((x, y) − (¯

x, ¯ y)) ≤ 0 separates ˆ x from Cx(ξ) and involves only the x variables. This hyperplane is the deepest valid cut that separates ˆ x from Cx(ξ), if depth is computed in Euclidean distance.

Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 20 / 29

An alternative cut derivation approach

Observation: One can show that the “KKT cut” we discussed is simply (ˆ x − ¯ x)T (x − ¯ x) ≤ 0. Is this always the case? Remember: min

(x,y)∈Cx,y(ξ)

1 2x − ˆ x2

x

(P)

Theorem: Projection-based separation

Let Cx,y(ξ) be a closed set such that Cx(ξ) = ProjxCx,y(ξ) is convex, and ˆ x ∈ Cx(ξ). Let (¯ x, ¯ y) be the optimal solution to (P). Then, the hyperplane: (ˆ x − ¯ x)T (x − ¯ x) ≤ 0 separates ˆ x from Cx(ξ), and is maximally violated in Euclidean distance.

Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 21 / 29

Analysis

Interpretation: The difficulty of separation is entirely determined by the difficulty of

• ptimizing over Cx,y(ξ).

This approach also works if we have integrality constraints! We do not need an explicit description of the convex hull: it is enough to have the usual description Cx,y(ξ) := {(x, y) ∈ Rn × Zp : gi(x, y) ≤ 0, i = 1, . . . , m}.

Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 22 / 29

1

The mid-term hydro scheduling problem

2

Problem decomposition and branch-and-cut

3

A separation routine

4

Finiteness

5

Computational experiments

Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 23 / 29

Finiteness

Linear case (scenario subproblems are polyhedra): [Luedtke, 2013] assumes that the set of separating inequalities for each scenario subproblem is finite (extreme points of the dual). It is not difficult to show that in this case the Branch-and-Cut terminates in finite time and returns the optimum.

Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 24 / 29

Finiteness

Linear case (scenario subproblems are polyhedra): [Luedtke, 2013] assumes that the set of separating inequalities for each scenario subproblem is finite (extreme points of the dual). It is not difficult to show that in this case the Branch-and-Cut terminates in finite time and returns the optimum. Nonlinear discrete case: The set of separating inequalities is infinite. The finiteness proof given by [Luedtke, 2013] does not work. But intuitively the algorithm should converge:

◮ We are computing strong (i.e. deepest possible) separating inequalities. ◮ How many of them can there possibly be?

Can we show that this algorithm converges even in the nonlinear discrete case?

Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 24 / 29

Finiteness

Kelley’s cutting plane algorithm: [Kelley, 1960] shows that the problem min{cx : g(x) ≤ 0, x ∈ X}, g convex, can be solved by a pure cutting plane method on a compact X. The proof relies on the generation of extreme supports for the feasible

• region. In this case, the algorithm converges to the optimum.

Adapting the proof:

Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 25 / 29

Finiteness

Kelley’s cutting plane algorithm: [Kelley, 1960] shows that the problem min{cx : g(x) ≤ 0, x ∈ X}, g convex, can be solved by a pure cutting plane method on a compact X. The proof relies on the generation of extreme supports for the feasible

• region. In this case, the algorithm converges to the optimum.

Adapting the proof: . . . actually there is very little to be done. We can apply the same idea. The proof is based on the fact that the projection cuts always cut off the current point ˆ x and never cut off Cx(ξ). If X is a compact, any infinite sequence of points ˆ x must eventually converge to a point that is in Cx(ξ).

Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 25 / 29

Final remarks

Summary: Scenario subproblems Linear Cuts Polytime Sep Finite Alg Linear [Luedtke, 2013] Y Y Y Linear integer Y N Y Nonlinear convex Y Y Y Nonlinear convex integer Y N Y

Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 26 / 29

1

The mid-term hydro scheduling problem

2

Problem decomposition and branch-and-cut

3

A separation routine

4

Finiteness

5

Computational experiments

Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 27 / 29

Work in progress!

Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 28 / 29

Bibliography

Baslis, G. C. and Bakirtzis, G. A. (2011). Mid-term stochastic scheduling of a price-maker hydro producer with pumped storage. IEEE Transactions on Power Systems, 26(4):1856–1865. Charnes, A. and Cooper, W. W. (1963). Deterministic equivalents for optimizing and satisficing under chance constraints. Operations Research, 11(1):18–39. Charnes, A., Cooper, W. W., and Symonds, G. H. (1958). Cost horizons and certainty equivalents: An approach to stochastic programming of heating oil. Management Science, 4(3):235–263. Codato, G. and Fischetti, M. (2006). Combinatorial benders’ cuts for mixed-integer linear programming. Operations Research, 54(4):756–766. Geoffrion, A. M. (1972). Generalized benders decomposition. Journal of Optimization Theory and Applications, 10(4):237–260. Kelley, J. E. (1960). The cutting-plane method for solving convex programs. Journal of the Society of Industrial and Applied Mathematics, 8(4):703–712. Luedtke, J. (2013). A branch-and-cut decomposition algorithm for solving chance-constrained mathematical programs with finite support. Mathematical Programming, pages 1–26. Online first. Prekopa, A. (1970). On probabilistic constrained programmming. In Kuhn, H. W., editor, Proceedings of the Princeton Symposium on Mathematical Programming, pages 113–138, Princeton, NJ. Princeton University Press. Enrico Malaguti (DEI - Unibo) Nonlinear discrete CCP for hydro scheduling January 7th, 2015 29 / 29