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MS&E 211 Quadratic Programming Ashish Goel

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A simple quadratic program Minimize (x 1 ) 2 Subject to: -x 1 + x 2 ≥ 3 -x 1 – x 2 ≥ -2

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A simple quadratic program Minimize (x 1 ) 2 Subject to: -x 1 + x 2 ≥ 3 -x 1 – x 2 ≥ -2 MOST OPTIMIZATION SOFTWARE HAS A QUADRATIC OR CONVEX OR NON-LINEAR SOLVER THAT CAN BE USED TO SOLVE MATHEMATICAL PROGRAMS WITH LINEAR CONSTRAINTS AND A MIN-QUADRATIC OBJECTIVE FUNCTION EASY IN PRACTICE

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A simple quadratic program Minimize (x 1 ) 2 Subject to: -x 1 + x 2 ≥ 3 -x 1 – x 2 ≥ -2 MOST OPTIMIZATION SOFTWARE HAS A QUADRATIC OR CONVEX OR NON-LINEAR SOLVER THAT CAN BE USED TO SOLVE MATHEMATICAL PROGRAMS WITH LINEAR CONSTRAINTS AND A MIN-QUADRATIC OBJECTIVE FUNCTION EASY IN PRACTICE QUADRATIC PROGRAM

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Next Steps Why are Quadratic programs (QPs) easy? Formal Definition of QPs Examples of QPs

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Next Steps Why are Quadratic programs (QPs) easy? – Intuition; not formal proof Formal Definition of QPs Examples of QPs – Regression and Portfolio Optimization

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Approximating the Quadratic Approximate x 2 by a set of tangent lines (here x is a scalar, corresponding to x 1 in the previous slides) d(x 2 )/dx = 2x, so the tangent line at (a, a 2 ) is given by y – a 2 = 2a (x-a) or y = 2ax – a 2 The upper envelope of the tangent lines gets closer and closer to the real curve

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Approximating the Quadratic Minimize Max {y 1, y 2, y 3, y 4, y 5, y 6, y 7 } Subject to: -x 1 + x 2 ≥ 3 -x 1 – x 2 ≥ -2 y 1 = 0 y 2 = 2x 1 – 1 y 3 = -2x 1 – 1 y 4 = 4x 1 – 4 y 5 = -4x 1 – 4 y 6 = x 1 – 0.25 y 7 = -x 1 – 0.25 Minimize (x 1 ) 2 Subject to: -x 1 + x 2 ≥ 3 -x 1 – x 2 ≥ -2

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Approximating the Quadratic Minimize z Subject to: -x 1 + x 2 ≥ 3 -x 1 – x 2 ≥ -2 z ≥ 0 z ≥ 2x 1 – 1 z ≥ -2x 1 – 1 z ≥ 4x 1 – 4 z ≥ -4x 1 – 4 z ≥ x 1 – 0.25 z ≥ -x 1 – 0.25 Minimize (x 1 ) 2 Subject to: -x 1 + x 2 ≥ 3 -x 1 – x 2 ≥ -2

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Approximating the Quadratic Minimize z Subject to: -x 1 + x 2 ≥ 3 -x 1 – x 2 ≥ -2 z ≥ 0 z ≥ 2x 1 – 1 z ≥ -2x 1 – 1 z ≥ 4x 1 – 4 z ≥ -4x 1 – 4 z ≥ x 1 – 0.25 z ≥ -x 1 – 0.25 Minimize (x 1 ) 2 Subject to: -x 1 + x 2 ≥ 3 -x 1 – x 2 ≥ -2 LPs can give successively better approximations

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Approximating the Quadratic Minimize z Subject to: -x 1 + x 2 ≥ 3 -x 1 – x 2 ≥ -2 z ≥ 0 z ≥ 2x 1 – 1 z ≥ -2x 1 – 1 z ≥ 4x 1 – 4 z ≥ -4x 1 – 4 z ≥ x 1 – 0.25 z ≥ -x 1 – 0.25 Minimize (x 1 ) 2 Subject to: -x 1 + x 2 ≥ 3 -x 1 – x 2 ≥ -2 Quadratic Programs = Linear Programs in the “limit”

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QPs and LPs Is it necessarily true for a QP that if an optimal solution exists and a BFS exists, then an optimal BFS exists?

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QPs and LPs Is it necessarily true for a QP that if an optimal solution exists and a BFS exists, then an optimal BFS exists? NO!! Intuition: When we think of a QP as being approximated by a succession of LPs, we have to add many new variables and constraints; the BFS of the new LP may not be the same as the BFS of the feasible region for the original constraints.

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QPs and LPs In any QP, it is still true that any local minimum is also a global minimum Is it still true that the average of two feasible solutions is also feasible?

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QPs and LPs In any QP, it is still true that any local minimum is also a global minimum Is it still true that the average of two feasible solutions is also feasible? – Yes!!

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QPs and LPs In any QP, it is still true that any local minimum is also a global minimum Is it still true that the average of two feasible solutions is also feasible? – Yes!! QPs still have enough nice structure that they are easy to solve

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Formal Definition of a QP Minimize c T x + y T y s.t. Ax = b Ex ≥ f Gx ≤ h y = Dx Where x, y are decision variables. All vectors are column vectors.

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Formal Definition of a QP Minimize c T x + y T y s.t. Ax = b Ex ≥ f Gx ≤ h y = Dx Where x, y are decision variables. All vectors are column vectors. The quadratic part is always non-negative

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Minimize c T x + y T y s.t. Ax = b Ex ≥ f Gx ≤ h y = Dx Where x, y are decision variables. All vectors are column vectors. Formal Definition of a QP i.e. ANY LINEAR CONSTRAINTS

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Equivalently Minimize c T x + (Dx) T (Dx) s.t. Ax = b Ex ≥ f Gx ≤ h Where x are decision variables. All vectors are column vectors.

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Equivalently Minimize c T x + x T D T Dx s.t. Ax = b Ex ≥ f Gx ≤ h Where x are decision variables. All vectors are column vectors.

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Equivalently Minimize c T x + x T Px s.t. Ax = b Ex ≥ f Gx ≤ h Where x are decision variables. All vectors are column vectors. P is positive semi-definite (a matrix that can be written as D T D for some D)

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Equivalently Minimize c T x + y T y s.t. Ax = b Ex ≥ f Gx ≤ h Where x are decision variables, and y represents a subset of the coordinates of x. All vectors are column vectors.

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Equivalently Instead of minimizing, the objective function is Maximize c T x – x T Px For some positive semi-definite matrix P

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Is this a QP? Minimize xy s.t. x + y = 5

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Is this a QP? Minimize xy s.t. x + y = 5 No, since x = 1, y=-1 gives xy = -1. Hence xy is not an acceptable quadratic part for the objective function.

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Is this a QP? Minimize xy s.t. x + y = 5 x, y ≥ 0

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Is this a QP? Minimize xy s.t. x + y = 5 x, y ≥ 0 No, for the same reason as before!

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Is this a QP? Minimize x 2 -2xy + y 2 - 2x s.t. x + y = 5

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Is this a QP? Minimize x 2 -2xy + y 2 - 2x s.t. x + y = 5 Yes, since we can write the quadratic part as (x- y)(x-y).

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A Useful Fact If P and Q are positive semi-definite, then so is P + Q

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An example: Linear Regression Let f be an unknown real-valued function defined on points in d dimensions. We are given the value of f on K points, x 1,x 2, …,x K, where each x i is d × 1 f(x i ) = y i Goal: Find the best linear estimator of f Linear estimator: Approximate f(x) as x T p + q – p and q are decision variables, (p is d × 1, q is scalar) Error of the linear estimator for x i is denoted Δ i Δ i = (x i ) T p + q - y i

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Linear Regression Best linear estimator: one which minimizes the error – Individual error for x i : Δ i – Overall error: commonly used formula is the sum of the squares of the individual errors

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Linear Least Squares Regression QP: Minimize Σ i (Δ i ) 2 s.t. For all i in {1..K}: Δ i = (x i ) T p + q - y i

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Linear Least Squares Regression QP: Minimize Σ i (Δ i ) 2 s.t. For all i in {1..K}: Δ i = (x i ) T p + q - y i Can simplify this further.

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Linear Least Squares Regression QP: Minimize Σ i (Δ i ) 2 s.t. For all i in {1..K}: Δ i = (x i ) T p + q - y i Can simplify this further. Let X denote the d × K matrix obtained from all the x i ’s: X = (x 1 x 2 … x K )

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Linear Least Squares Regression QP: Minimize Σ i (Δ i ) 2 s.t. For all i in {1..K}: Δ i = (x i ) T p + q - y i Can simplify this further. Let X denote the d × K matrix obtained from all the x i ’s: X = (x 1 x 2 … x K ) Let e denote a K × 1 vector of all 1’s

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Linear Least Squares Regression QP: Minimize Δ T Δ s.t. Δ = X T p + qe – y

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Simple Portfolio Optimization Consider a market with N financial products (stocks, bonds, currencies, etc.) and M future market scenarios Payoff matrix P: P i,j = Payoff from product j in the i-th scenario x j = # of units bought of j-th product c j = cost per unit of j-th product Additional assumption: Probability q i of market scenario i happening is given

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Simple Portfolio Optimization Example: Stock mutual fund and bond mutual fund, each costing $1, with two scenarios, occurring with 50% probability each: that the economy will grow next year or stagnate PAYOFF MATRIX STOCKBOND GROWTH0.30.05 STAGNATION-0.10.05

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Simple Portfolio Optimization Example: Stock mutual fund and bond mutual fund, each costing $1, with two scenarios, occurring with 50% probability each: that the economy will grow next year or stagnate PAYOFF MATRIX STOCKBOND GROWTH0.30.05 STAGNATION-0.10.05 What portfolio maximizes expected payoff? 100% STOCK, 50% EACH, 100% BOND

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Simple Portfolio Optimization Example: Stock mutual fund and bond mutual fund, each costing $1, with two scenarios, occurring with 50% probability each: that the economy will grow next year or stagnate PAYOFF MATRIX STOCKBOND GROWTH0.30.05 STAGNATION-0.10.05 What portfolio maximizes expected payoff? 100% STOCK, 50% EACH, 100% BOND

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Simple Portfolio Optimization Example: Stock mutual fund and bond mutual fund, each costing $1, with two scenarios, occurring with 50% probability each: that the economy will grow next year or stagnate PAYOFF MATRIX STOCKBOND GROWTH0.30.05 STAGNATION-0.10.05 What portfolio minimizes variance? 100% STOCK, 50% EACH, 100% BOND

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Simple Portfolio Optimization Example: Stock mutual fund and bond mutual fund, each costing $1, with two scenarios, occurring with 50% probability each: that the economy will grow next year or stagnate PAYOFF MATRIX STOCKBOND GROWTH0.30.05 STAGNATION-0.10.05 What portfolio minimizes variance? 100% STOCK, 50% EACH, 100% BOND

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Simple Portfolio Optimization Example: Stock mutual fund and bond mutual fund, each costing $1, with two scenarios, occurring with 50% probability each: that the economy will grow next year or stagnate PAYOFF MATRIX STOCKBOND GROWTH0.30.05 STAGNATION-0.10.05 What portfolio minimizes variance subject to getting at least 7.5% expected returns? 100% STOCK, 50% EACH, 100% BOND

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Simple Portfolio Optimization Example: Stock mutual fund and bond mutual fund, each costing $1, with two scenarios, occurring with 50% probability each: that the economy will grow next year or stagnate PAYOFF MATRIX STOCKBOND GROWTH0.30.05 STAGNATION-0.10.05 What portfolio minimizes variance subject to getting at least 7.5% expected returns? 100% STOCK, 50% EACH, 100% BOND

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Minimizing Variance (≈ Risk) Often, we want to minimize the variance of our portfolio, subject to some cost budget b and some payoff target π Let y i denote the payoff in market scenario i y i = P i x Expected payoff= z = Σ i q i y i = q T y Variance = Σ i q i (y i - z) 2 = Σ i ((q i ) 1/2 (y i - z)) 2 Let v i denote (q i ) 1/2 (y i – z)

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Portfolio Optimization: QP Minimize v T v s.t. c T x ≤ b y = Px z = q T y z ≥ π (for all i in {1…K}): v i = (q i ) 1/2 (y i – z)

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Chapter 3 Linear Programming Methods

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