Our first lecture was about the important and powerful technique of

One person asked why, in a line of the proof of Theorem 1.3, we can use $\inf_{c\in \mathbb{R}^m} \inf_{x\in X(c)} =\inf_{x\in X}$. Another class member provided the correct answer: that every $x\in X$ satisfies $h(x)=c$ for some $c$.

**Lagrange multipliers**, which are used to address problems of optimization under constraints. Additional reading can be found in S. Boyd and L. Vandenberghe: Convex Optimization. Cambridge University Press (2004), chapters 1 and 5.One person asked why, in a line of the proof of Theorem 1.3, we can use $\inf_{c\in \mathbb{R}^m} \inf_{x\in X(c)} =\inf_{x\in X}$. Another class member provided the correct answer: that every $x\in X$ satisfies $h(x)=c$ for some $c$.

Subsequent to the lecture I corrected the numbering of Theorem 1.3 on page 3. I will sometimes make small changes to the notes after a lecture.

As we begin, some of you may be wondering: what is Operational Research? Here are some links that may help to explain.

**PROFESSIONAL SOCIETIES**

INFORMS is US-based,society but has members throughout the world and organizes many conferences. They have sections in theoretical fields such as Applied Probability, and more applied ones such as Transportation Science & Logistics. On their web site they try answer: What is Operations Research?

They say, "Operations Research (O.R.), or operational research in the U.K, is a discipline that deals with the application of advanced analytical methods to help make better decisions. The terms management science and analytics are sometimes used as synonyms for operations research. Employing techniques from other mathematical sciences, such as mathematical modeling, statistical analysis, and mathematical optimization, operations research arrives at optimal or near-optimal solutions to complex decision-making problems."

The UK society is the the Operational Research Society: https://www.theorsociety.com/

**RESEARCH FUNDING**

Here is a graphic of research areas in the Mathematical Sciences supported by EPSRC

Total theme funding, £210.2 million (4.66% of whole portfolio). There are 350 grants in the Mathematical Sciences theme. Mathematical Aspects of Operational Research presently has 21 grants, totalling £8m and is represented by the circle at the bottom of the picture.

**RESEARCH PUBLICATION**

Mathematics of Operations Research is a leading journal. If you look at the titles of some articles this will show that there is some quite deep mathematics being pursued.

Mathematics of Operations Research (MOR) publishes excellent foundational articles having significant mathematical contents and relevance to operations research and management science.