The rise of highly mathematical quantitative methods in investment and finance in the past 20 years has exposed the fact that there is a considerable amount of pseudomathematical and pseudoscientific nonsense in the field. As a result, individual investors and fund managers alike are often misled as to the true level of risk in proposed investments. For example, many new investment strategies and exchange-traded funds are promoted by citing "backtest" results -- simulations of a strategy's performance based on historical market data. But what is almost never disclosed is how many different variations of the strategy were explored in developing the strategy. Yet, as myself and three co-authors have shown in our recent paper "Pseudo-mathematics and financial charlatanism: The effects of backtest overfitting on out-of-sample performance," if enough variations of a strategy are tried, almost any desired Sharpe ratio (a standard measure of portfolio performance) can be achieved, due to statistical chance alone. Indeed, it is now thought that statistical overfitting of backtest data may be the principal reason that many new investment strategies and exchange-traded funds, which look good based on published backtests, nonetheless fall flat when actually fielded in the market.

All of this is closely related to the larger issue of reproducibility and professional ethics in the field of scientific research. Just as there is a movement now to require pharmaceutical companies to disclose all results of field tests, good and bad, so researchers and investors should require full disclosure of the extent of backtesting involved in the development of an investment strategy or fund. In a larger sense, we need to ask ourselves why so few are willing to speak out when questionable methodologies and claims are seen in the field of finance (or any other field). As we wrote in our recent paper, "Our silence is consent, making us accomplices in these abuses."

In this talk, we introduce a certain polytope, H, induced by a particular discounted Markov decision process corresponding to a given graph G. It has been proved that if the graph G is Hamiltonian, then corresponding to each Hamiltonian cycle in graph G, there exists an extreme point in polytope H, namely, Hamiltonian extreme point. We present our theoretical results on the geometric properties of these extreme points and show that by exploiting them, two new counterpart problems for the Hamiltonian Cycle Problem can be constructed : (1) Searching for one Hamiltonian extreme point in the polytope H; (2) Finding the volume of the polytope H with a desired level of precision. At the end, we conclude this talk with two conjectures derived from these new counterpart problems and provide supporting numerical results.

We give precise lower bounds on the number of edges of an arbitrary d-dimensional polytope with v vertices, for an interesting range of values of v and d. Previously this problem had only been studied in detail for simplicial polytopes. If time permits, we may indicate the connection with quasilinear mappings on not necessarily locally convex spaces.

We propose a version of bundle method for minimizing locally Lipschitz and prox-regular functions. A nonmonotone trust region method with box trust region is used so that we solve a linear programming problem in each iteration. We adopt a convexification technique so that adding on a dynamical quadratic term can make the objective function convexifiable. Global convergence is showed in the sense that the iteration sequence converges to a fixed point of the proximal point mapping. This talk is based a PhD project in RMIT.

In this talk the turnpike property is investigated for convex optimal control problems with undiscounted integral functionals. The system is described by a multi-valued mapping having convex graph, and the utility function is assumed to be concave but not necessarily strictly concave. Turnpike theorems are proved under the main assumption that over any given line segment, either the multi-valued mapping is strictly convex or the utility function is strictly concave.

Risk management method based on the growth portfolio theory is theoretically superior but often too risky in practice. We will illustrate both in theory and by simulation that the disconnection is due to that in practice investors only invest for a finite time horizon and they need to explicitly address the risk. Incorporate these practical consideration we derive more realistic conservative capital allocation estimate for risk management. Nonsmooth risk functions naturally arise in such problems providing an interesting application example of variational analysis.

This talk is based on collaborative research with Ralph Vince (LSP Partners, LLC) and Marcos Lopez de Prado (Guggenheim Partners).

Several local regularity properties of finite collections of sets will be discussed, namely: semiregularity, subregularity and uniform regularity. Metric and dual quantitative characterizations of these properties will be provided as well as their relationships with the corresponding properties of set-valued mappings.

The Pataki sandwich theorem provides (different) necessary and sufficient conditions for a closed convex cone to be facially dual complete (or nice). Examples demonstrate that both conditions are not sharp. We tighten the necessary condition (facial exposedness) further, hence sharpening the sandwich theorem. This is joint work with Levent Tuncel, University of Waterloo.

In this talk, I present some recent algorithms for non-smooth and non-convex optimisation problems. More specifically, I will demonstrate the secant method, the quasi-secant method, a generalised subgradient method, and an approximate subgradient algorithm for unconstrained problems. The theoretical backgrounds of these methods are explained and some supportive numerical results are demonstrated.

We present a measure-theoretic foundation for analysing the expectation of a smooth complex-valued function defined over the attractor of an Iterated Function System (IFS). A Chaos-Game algorithm for numerically computing such expectations naturally follows. This work extends the results of our previous paper (with David Bailey and Richard Crandall), in which such expectations were defined over a restricted class of string-generated Cantor sets, motivated by the desire to analyse neural spatial distributions.

The justly celebrated von Neumann minimax theorem has many proofs. Here I reproduce the most complex one I am aware of. This provides a fine didactic example for many courses in convex analysis or functional analysis.