**Robust Optimization**

and Applications

**Thanks**

**Optimization models**

**Pitfalls**

**Robust Optimization Paradigm**

**Approximating a robust
solution**

**Agenda**

**LP as a conic problem**

**Second-order cone
programming**

**Semidefinite programming**

**Dual form of conic program**

**Robust conic programming**

**Polytopic uncertainty**

**Robust LP**

**Robust LP with ellipsoidal
uncertainty**

**Robust LP as SOCP**

**Example: robust portfolio
design**

**Solution of robust portfolio
problem**

**Robust SOCP**

**Example: robust
least-squares**

**Robust SDP**

**Example: robust control**

**Analysis of robust conic
problems**

**Relaxations**

**Quality estimates**

**Quality estimates: some
results**

**restriction**

**Sampling**

**Variations on Robust Conic
Programming**

**A Boolean problem**

**Max-quad as a robust LP**

**Rank relaxation**

**Boolean optimization:
geometric approach**

**SDP for boolean / nonconvex
optimization**

**Robust boolean optimization**

**SDP relaxation of robust
problem**

**Chance-constrained
programming**

**Problems with adjustable
parameters**

**Adjustable parameters: some
results**

**Link with feedback control**

**Challenges**

**Set estimation**

**Part I: summary**

**Slide 44**

**Part II: Contextual
Applications**

**Robust path planning**

**Uncertainty in Markov
Decision Process**

**Agenda**

**Markov decision problem**

**Previous Work**

**Robust dynamic programming**

**Inner problem**

**Worst-case performance of a
policy**

**Describing uncertainty**

**Joint estimation and
optimization**

**Estimating a transition
matrix**

**Likelihood regions**

**likelihood regions**

**Reduction to a 1-D problem**

**Complexity results**

**Application to aircraft
routing**

**Markov chain model for the
storms**

**information update and recourse**

**Dynamic programming model**

**Nominal algorithm**

**Sample path planning**

**Improvements over obvious
strategies**

**Robustness**

**Optimality vs. uncertainty
level**

**Errors in uncertainty level**

**Extensions**

**Summary of results**

**Some references**

**Robust Classification**

**Linear Classification**

**What is a classifier?**

**Classification constraints**

**robust classification:
**

support vector machine

**box uncertainty model**

**formulations**

**extensions**

**minimax probability machine**

**Problem statement**

**SOCP formulation**

**Dual problem**

**Geometric interpretation**

**Robust classification:
**

summary of results

**Wrap-up**