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Ellipsoid-type Confidential Bounds on Semi-algebraic Sets via SDP Relaxation

Makoto YamashitaMasakazu Kojima Tokyo Institute of Technology

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Confidential Bounds in Polynomial Optimization Problem

minOptimal

SDP relaxation(convex region)

SDP solution

Local adjustmentfor feasible region

We compute this ellipsoid by SDP. Optimal solutions exist in this ellipsoid.

Feasible region

Semi-algebraic Sets

(Polynomials)

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Ellipsoid research .

MVEE (the minimum volume enclosing ellipsoid)

Our approach based on SDP relaxation

Solvable by SDP Small computation cost

⇒We can execute multiple times changing

Outline

1. Math Form of Ellipsoids2. SDP relaxation3. Examples of POP4. Tightness of Ellipsoids

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Mathematical Formulation . Ellipsoid

We define

.

By some steps, we consider SDP relaxation

.

.

Note that Furthermore

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Lifting⇒

quadraticlinear (easier)

Still difficult

(convex hull)

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SDP relaxation . .

relaxation

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. .

Gradient Optimal attained at

.

Cover

Inner minimization

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Relations of

SDP

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Example from POP ex9_1_2 from GLOBAL library

(http://www.gamsworld.org/global/global.htm)

We use SparsePOP to solve this by SDP relaxationSparsePOPhttp://sparsepop.sourceforge.net

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Region of the Solution

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Reduced POP

Optimal Solutions:

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Ellipsoids for Reduced SDP

Optimal Solutions:

Very tight bound

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Results on POP

Very good objective values ex_9_1_2 & ex_9_1_8 have multiple optimal

solutions ⇒ large radius

Tightness of Ellipsoids Target set

6 Shape Matricies

We draw 2D picture,

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The case p=2 (2 constraints)

The ellipsoids are tight.

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Target set

6 ellipsoids by SDP

More constraints

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Ellipsoids shrink.But its speed is slower than the target set.

p=2p=32

p=128

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Conclusion & Future works An enclosing ellipsoid by SDP relaxation

Improve the SDP solution of POP Very low computation cost

Successive ellipsoid for POP sometimes stops before bounding the region appropriately

Ellipsoids may become loose in the case of many constraints

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Thank you very much for your attention.

This talk is based on the following technical paperMasakazu Kojima and Makoto Yamashita,“Enclosing Ellipsoids and Elliptic Cylinders of Semialgebraic Sets and Their Application to Error Boundsin Polynomial Optimization”, Research Report B-459, Dept. of Math. and Comp. Sciences,Tokyo Institute of Technology, Oh-Okayama, Meguro, Tokyo 152-8552,January 2010.