Structured Sparsity in Polynomial Optimization: Sparse SDP Relaxation and PDE Applications, Slides of Mathematics

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Sparsity in Polynomial Optimization

Contents

1. How do we formulate structured sparsity? 1-1. Unconstrained cases. 1-2. Constrained and linear objective function cases. (Recent Joint work with S.Kim & K.Kobayashi)
2. Sparse SDP relaxation of constrained POPs.
3. Applications to PDEs (partial differential equations).
4. Concluding remarks. POP --- Polynomial Optimization Problem

Contents

1. How do we formulate structured sparsity? 1-1. Unconstrained cases. 1-2. Constrained and linear objective function cases.
2. Sparse SDP relaxation of constrained POPs.
3. Applications to PDEs (partial differential equations).
4. Concluding remarks. POP --- Polynomial Optimization Problem

Contents

1. How do we formulate structured sparsity? 1-1. Unconstrained cases. 1-2. Constrained and linear objective function cases.
2. Sparse SDP relaxation of constrained POPs.
3. Applications to PDEs (partial differential equations).
4. Concluding remarks.
   • We consider cases where objective functions are linear.
   • LP, SOCP and SDP + Primal-Dual Interior-Point Method.

Example csp matrix R = (n=20)

Example

Contents

1. How do we formulate structured sparsity? 1-1. Unconstrained cases. 1-2. Constrained and linear objective function cases.
2. Sparse SDP relaxation of constrained POPs. (Joint work with S.Kim, M.Muramatsu & H.Waki)
3. Applications to PDEs (partial differential equations).
4. Concluding remarks. Sections 1-1 + 1-2 ==> Section 2 Sparse SDP relaxation = Modification of Lasserre’s relaxation

Contents

1. How do we formulate structured sparsity? 1-1. Unconstrained cases. 1-2. Constrained and linear objective function cases.
2. Sparse SDP relaxation of constrained POPs.
3. Applications to PDEs (partial differential equations). (Ongoing joint work with M.Mevissen, J.Nie & N.Takayama)
4. Concluding remarks.