\begin{align} \min_{x}\quad & f_{0}\qty(x) \\ \textrm{s.t.} \quad & f_{i}\qty(x) \leq 0, i = 1 \dots m\\ &Ax = b \end{align}

Where \(f_{0}\) is quasiconvex, and \(f_{i\geq 1}\) is convex.

solution methods

convex representation of sublevel sets

if \(f_{0}\) is quasiconvex, then there exists a family of functions \(\phi_{t}\) for which:

\begin{equation} f_{0} \leq t \iff \phi_{t}\qty(x) \leq 0 \end{equation}

where \(\phi_{t}\) is convex for fixed \(t\).

bisection method for quasiconvex optimization

We can cast all quasiconvex problems into a binary search over \(t\). Solve the following convex feasibility problem:

Key Sequence

Notation

New Concepts

• Strong and Weak Duality
• Geometric Interperattion of the Dual
  • Complementary Slackness
  • KKT Conditions
• preturbation analysis
• dual transformation
• theorem of alternatives

Important Results / Claims

Questions

Interesting Factoids

Key Sequence

Notation

New Concepts

• Duality for Feasible Problems
• Approximation and Fitting
  • Huber Penalty Function

Important Results / Claims

• Farkas’ Lemma

Questions

Interesting Factoids

Key Sequence

Notation

New Concepts

• Lagrangian
• Lagrange Dual Problem
  • Strong and Weak Duality
  • constraint qualifications

Important Results / Claims

• Connection between duals and conjugates

Questions

Interesting Factoids

Revenue vs Market Cap

• large Chinese coporations are not listed on stocks or only a bit of their shares
• thus revenue / market cap is very different measures

Role of State-Owned Enterprises

1. almost 40 years after market reforms, China’s state-owned enterprises still have a large role
2. state owned enterprises are called on to support economic growth for China/deliver aid/satisfy industrial policies

Problems with SOEs

• lots of debt
• pushed down future productive growth and pushes up public debt
• current policies not tackling SOE performance

Private firms is contributing to about half of economic exports