Two Exercices about Stochastic Gradient Option Pricing Problem and Variance Reduction Spatial Rendez-vous Under Probability Constraint Applications of the Stochastic Gradient Method December 11, 2019 P. Carpentier Master Optimization — Stochastic Optimization 2019-2020 149 / 328
Two Exercices about Stochastic Gradient Option Pricing Problem and Variance Reduction Spatial Rendez-vous Under Probability Constraint Lecture Outline Two Exercices about Stochastic Gradient 1 Two-Stage Recourse Problem Trade-off Between Investment and Operation Option Pricing Problem and Variance Reduction 2 Pricing Problem Modeling Computing Efficiently the Price Spatial Rendez-vous Under Probability Constraint 3 Satellite Model and Optimization Problem Probability and Conditional Expectation Handling Stochastic APP Algorithm Numerical Results P. Carpentier Master Optimization — Stochastic Optimization 2019-2020 150 / 328
Two Exercices about Stochastic Gradient Two-Stage Recourse Problem Option Pricing Problem and Variance Reduction Trade-off Between Investment and Operation Spatial Rendez-vous Under Probability Constraint Two Exercices about Stochastic Gradient 1 Two-Stage Recourse Problem Trade-off Between Investment and Operation Option Pricing Problem and Variance Reduction 2 Pricing Problem Modeling Computing Efficiently the Price Spatial Rendez-vous Under Probability Constraint 3 Satellite Model and Optimization Problem Probability and Conditional Expectation Handling Stochastic APP Algorithm Numerical Results P. Carpentier Master Optimization — Stochastic Optimization 2019-2020 151 / 328
Two Exercices about Stochastic Gradient Two-Stage Recourse Problem Option Pricing Problem and Variance Reduction Trade-off Between Investment and Operation Spatial Rendez-vous Under Probability Constraint Two Exercices about Stochastic Gradient 1 Two-Stage Recourse Problem Trade-off Between Investment and Operation Option Pricing Problem and Variance Reduction 2 Pricing Problem Modeling Computing Efficiently the Price Spatial Rendez-vous Under Probability Constraint 3 Satellite Model and Optimization Problem Probability and Conditional Expectation Handling Stochastic APP Algorithm Numerical Results P. Carpentier Master Optimization — Stochastic Optimization 2019-2020 152 / 328
Two Exercices about Stochastic Gradient Two-Stage Recourse Problem Option Pricing Problem and Variance Reduction Trade-off Between Investment and Operation Spatial Rendez-vous Under Probability Constraint A Basic Two-Stage Recourse Problem We consider the management of a water reservoir. Water is drawn from the reservoir by way of random consumers. In order to ensure the water supply, 2 decisions are taken at 2 successive time steps. A first supply decision q 1 is taken without any knowledge of the effective consumption, the associated cost being equal � � 2 , with c 1 > 0. to c 1 q 1 Once the consumption d (realization of a random variable D ) has been observed, a second supply decision q 2 is taken in order to maintain the reservoir at its initial level, that is, q 2 = d − q 1 . � � 2 , with c 2 > 0. The associated cost is equal to c 2 q 2 The problem is to minimize the expected overall cost of operation. P. Carpentier Master Optimization — Stochastic Optimization 2019-2020 153 / 328
Two Exercices about Stochastic Gradient Two-Stage Recourse Problem Option Pricing Problem and Variance Reduction Trade-off Between Investment and Operation Spatial Rendez-vous Under Probability Constraint Mathematical Formulation and Solution Problem Formulation q 1 is a deterministic decision variable, whereas q 2 is the realization of a random variable Q 2 . � � 2 � � � 2 + E � ( q 1 , Q 2) c 1 min q 1 c 2 Q 2 s.t. q 1 + Q 2 = D P -a.s. . Equivalent Problem � � 2 � � � 2 + c 2 � min c 1 q 1 D − q 1 q 1 ∈ R E � � c 2 Analytical solution : q ♯ 1 = E D . c 1 + c 2 P. Carpentier Master Optimization — Stochastic Optimization 2019-2020 154 / 328
Two Exercices about Stochastic Gradient Two-Stage Recourse Problem Option Pricing Problem and Variance Reduction Trade-off Between Investment and Operation Spatial Rendez-vous Under Probability Constraint Stochastic Gradient Algorithm � − 2 c 2 D ( k +1) � α Q ( k +1) = Q ( k ) 2( c 1 + c 2 ) Q ( k ) − . 1 1 1 k + β Algorithm (initialization) Algorithm (iterations) // // // Random generator // Algorithm // // rand(’normal’); rand(’seed’,123); q1k = 10.; // for k = 1:100 // Random consumption dk = m + (sd*rand(1)); // gk = 2*((c1+c2)*q1k) - 2*(c2*dk); m = 10.; sd = 5.; epsk = 1/(k+10); // q1k = q1k - (epsk*gk); // Criterion x = [x ; k]; y = [y ; q1k]; // end c1 = 3.; c2 = 1.; // // // Trajectory plot // Initialization // // plot2d(x,y); x = [ ]; y = [ ]; xtitle(’Stochastic Gradient ’,’Iter.’,’q1’); P. Carpentier Master Optimization — Stochastic Optimization 2019-2020 155 / 328
Two Exercices about Stochastic Gradient Two-Stage Recourse Problem Option Pricing Problem and Variance Reduction Trade-off Between Investment and Operation Spatial Rendez-vous Under Probability Constraint A Realization of the Algorithm P. Carpentier Master Optimization — Stochastic Optimization 2019-2020 156 / 328
Two Exercices about Stochastic Gradient Two-Stage Recourse Problem Option Pricing Problem and Variance Reduction Trade-off Between Investment and Operation Spatial Rendez-vous Under Probability Constraint More Realizations P. Carpentier Master Optimization — Stochastic Optimization 2019-2020 157 / 328
Two Exercices about Stochastic Gradient Two-Stage Recourse Problem Option Pricing Problem and Variance Reduction Trade-off Between Investment and Operation Spatial Rendez-vous Under Probability Constraint Slight Modification of the Problem As in the basic two-stage recourse problem, a first supply decision q 1 is taken without any knowledge of the effective consumption, the associated cost being equal to � � 2 , c 1 q 1 a second supply decision q 2 is taken once the consumption d has been observed (realization of a r.v. D ), the cost of this � � 2 . second decision being equal to c 2 q 2 The difference between supply and demand is penalized thanks to � � 2 . The new problem is : an additional cost term c 3 q 1 + q 2 − d � � 2 � � � 2 + c 2 � � 2 + c 3 � min q 1 + Q 2 − D . ( q 1 , Q 2) E c 1 q 1 Q 2 Question: how to solve it using a stochastic gradient algorithm? P. Carpentier Master Optimization — Stochastic Optimization 2019-2020 158 / 328
Two Exercices about Stochastic Gradient Two-Stage Recourse Problem Option Pricing Problem and Variance Reduction Trade-off Between Investment and Operation Spatial Rendez-vous Under Probability Constraint Resolution of the Modified Problem Idea : use the interchange theorem to solve the problem w.r.t. Q 2 . � � � 2 + c 2 � � 2 + c 3 � � 2 � q 1 + Q 2 − D ( q 1 , Q 2) E min c 1 q 1 Q 2 � � 2 �� � � 2 + min � � � 2 + c 3 � ⇐ ⇒ min c 1 q 1 c 2 Q 2 q 1 + Q 2 − D E q 1 Q 2 � � 2 �� � � 2 + E � � � 2 + c 3 � ⇐ ⇒ min q 1 + q 2 − D c 1 q 1 min q 2 c 2 q 2 . q 1 The optimal solution of the minimization problem w.r.t. q 2 is � � c 3 ♯ = Q 2 D − q 1 c 2 + c 3 so that the problem is equivalent to the open-loop problem � � � � 2 + � � 2 c 2 c 3 min c 1 q 1 q 1 − D . q 1 E c 2 + c 3 The stochastic gradient algorithm applies! P. Carpentier Master Optimization — Stochastic Optimization 2019-2020 159 / 328
Two Exercices about Stochastic Gradient Two-Stage Recourse Problem Option Pricing Problem and Variance Reduction Trade-off Between Investment and Operation Spatial Rendez-vous Under Probability Constraint Two Exercices about Stochastic Gradient 1 Two-Stage Recourse Problem Trade-off Between Investment and Operation Option Pricing Problem and Variance Reduction 2 Pricing Problem Modeling Computing Efficiently the Price Spatial Rendez-vous Under Probability Constraint 3 Satellite Model and Optimization Problem Probability and Conditional Expectation Handling Stochastic APP Algorithm Numerical Results P. Carpentier Master Optimization — Stochastic Optimization 2019-2020 160 / 328
Two Exercices about Stochastic Gradient Two-Stage Recourse Problem Option Pricing Problem and Variance Reduction Trade-off Between Investment and Operation Spatial Rendez-vous Under Probability Constraint Trade-off Investment/Operation – Problem Statement A company owns N production units and has to meet a given non stochastic demand d ∈ R . For each unit i , the decision maker first takes an investment decision u i ∈ R , the associated cost being I i ( u i ). Then a discrete disturbance w i ∈ { w i , a , w i , b , w i , c } occurs. Knowing all noises, the decision maker selects for each unit i an operating point v i ∈ R , which leads to a cost C i ( u i , v i , w i ) and a production P i ( v i , w i ). The goal is to minimize the expected overall cost, subject to the following constraints: investment limitation: Θ( u 1 , . . . , u N ) ≤ 0, operation limitation: v i ≤ ϕ i ( u i ) , i = 1 . . . , N , demand satisfaction: � N i =1 P i ( v i , w i ) − d = 0. P. Carpentier Master Optimization — Stochastic Optimization 2019-2020 161 / 328
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