MTH424: Convex Analysis (Spring 2025)

Convex analysis is a branch of mathematics that studies convex sets and convex functions. A set is convex if a straight line between any two points in the set always stays inside it. This field is important in optimization, economics, and engineering. It helps in solving real-world problems like minimizing costs, maximizing profits, and designing efficient systems. Convex analysis is widely used in machine learning, finance, and physics. 😊

Learn the basic concepts of convex sets and convex functions.

Study the differential properties of convex functions.

Understand these inequalities and their real-world applications.

Improve logical thinking through homework and projects.

Gain confidence in solving mathematical problems independently.

This course will strengthen your mathematical foundation and problem-solving abilities! 😊

Convex sets and their properties, Convex hull and their properties, Best approximation theorem. Convex functions, Basic definitions, properties, various generalizations, Differentiable convex functions, Hermite and Hadamard inequalities, Subgradient, Characterizations and applications in linear and nonlinear optimization.

• Assignment 1 | Download PDF | View Online
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• Assignment 3 | Download PDF | View Online
• Assignment 4 | Download PDF | View Online

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1. If $f:I\to \mathbb{R}$ is convex, then $g(x):=\frac{f(x)-f(a)}{x-a}$, where $x>a$ is an increasing function.
2. If $f_1$ and $f_2$ are convex on $I$, then $f_1+f_2$ is convex on $I$. Also prove that $af$, where $a>0$ is convex if $f$ is convex.
3. Prove that if $f$ is differentiable and convex on $I$, then $f'$ is increasing and if $f$ is twice differentiable and convex on $I$ then $f'' (x)\geq 0$ for all $x\in I$.
4. Let $f:I\rightarrow\ \mathbb{R}$ and $g:J\rightarrow\mathbb{R}$, where $range(f)\subseteq J$. If $f$, $g$ are convex and $g$ is increasing, then the composite function $g\circ\ f$ is convex on $I$.
5. Let $f_i:I\rightarrow\ \mathbb{R}$ be an arbitrary family of convex functions and let $f(x)={\mathrm{sup}}_i f_i(x)$. If $J=[x\in I∶ f(x) <\infty]$ is non-empty then $J$ is an interval and $f$ is convex on $J$.
6. If $f_n:I\rightarrow\mathbb{R}$ is a sequence of convex functions converging to a finite limit function $f$ on $I$, then $f$ is convex on $I$.
7. If $f:I\rightarrow\mathbb{R}$ and $g:I\rightarrow\mathbb{R}$ are both non-negative, decreasing and convex functions, then $h(x)=f(x)g(x)$ is also non-negative, decreasing and convex on $I$.
8. Define log-convex function and prove that if $f$ and $g$ are two log-convex functions on $I$, then $fg$ is also convex on $I$.
9. Let $f$ be a convex function on interval $(a,b)$ and let $x_i\in (a,b)$ and $\alpha_i >0$ for $i=1,2,...,n$ such that $\sum_{i=1}^{n}\alpha_i=1$, then $f\left(\sum_{i=1}^{n}\alpha_i x_i \right)\leq \sum_{i=1}^{n} \alpha_i f(x_i)$.
10. State Hermite-Hadamard inequality and give its graphics illustration.

• http://en.wikipedia.org/wiki/Convex_function
• http://mathworld.wolfram.com/ConvexFunction.html

1. Roberts, A. W., & Varberg, D. E. (1973). Convex functions. Academic Press. (Google Book Preview)
2. Rockafellar, R. T. (1970). Convex analysis. Princeton University Press. (Google Book Preview)
3. Bauschke, H. H., & Combettes, P. L. (2011). Convex analysis and monotone operator theory in Hilbert spaces. Springer.
4. Bazaraa, M. S., Sherali, H. D., & Shetty, C. M. (2006). Nonlinear programming: Theory and algorithms (3rd ed.). Wiley-Interscience.
5. Niculescu, C. P., & Persson, L. E. (2006). Convex functions and their applications: A contemporary approach. Springer.