Jianjun Gao

Optimal multi-period mean–variance policy under no-shorting constraint

We consider in this paper the mean–variance formulation in multi-period portfolio selection under no-shorting constraint. Recognizing the structure of a piecewise quadratic value function, we prove that the optimal portfolio policy is piecewise linear with respect to the current wealth level, and derive the semi-analytical expression of the piecewise quadratic value function. One prominent feature of our findings is the identification of a deterministic time-varying threshold for the wealth process and its implications for market settings. We also generalize our results in the mean–variance formulation to utility maximization with no-shorting constraint.

Optimal Cardinality Constrained Portfolio Selection

One long-standing challenge in both the optimization and investment communities is to devise an efficient algorithm to select a small number of assets from an asset pool such that a portfolio objective is optimized. This cardinality constrained investment situation naturally arises due to the presence of various forms of market friction, such as transaction costs and management fees, or even due to the consideration of mental cost. Unfortunately, the combinatorial nature of such a portfolio selection problem formulation makes the exact solution process NP-hard in general. We focus in this paper on the cardinality constrained mean-variance portfolio selection problem. Instead of tailoring such a difficult problem into the general solution framework of mixed-integer programming formulation, we explore the special structures and rich geometric properties behind the mathematical formulation. Applying the Lagrangian relaxation to the primal problem results in a pure cardinality constrained portfolio selection ...

Cardinality Constrained Linear-Quadratic Optimal Control

As control implementation often incurs not only a variable cost associated with the magnitude or energy of the control, but also a setup cost, we consider a discrete-time linear-quadratic (LQ) optimal control problem with a limited number of control implementations, termed in this technical note the cardinality constrained linear-quadratic optimal control (CCLQ). We first derive a semi-analytical feedback policy for CCLQ problems using dynamic programming (DP). Due to the exponential growth of the complexity in calculating the action regions, however, DP procedure is only efficient for CCLQ problems with a scalar state space. Recognizing this fact, we develop then two lower-bounding schemes and integrate them into a branch-and-bound (BnB) solution framework to offer an efficient algorithm in solving general CCLQ problems. Adopting the devised solution algorithm for CCLQ problems, we can solve efficiently the linear-quadratic optimal control problem with setup costs.

Dynamic mean-risk portfolio selection with multiple risk measures in continuous-time

While our society began to recognize the importance to balance the risk performance under different risk measures, the existing literature has confined its research work only under a static mean-risk framework. This paper represents the first attempt to incorporate multiple risk measures into dynamic portfolio selection. More specifically, we investigate the dynamic mean-variance-CVaR (Conditional value at Risk) formulation and the dynamic mean-variance-SFP (Safety-First Principle) formulation in a continuous-time setting, and derive the analytical solutions for both problems. Combining a downside risk measure with the variance (the second order central moment) in a dynamic mean-risk portfolio selection model helps investors control both a symmetric central risk measure and an asymmetric catastrophic downside risk. We find that the optimal portfolio policy derived from our mean-multiple risk portfolio optimization models exhibits a feature of curved V-shape. Our numerical experiments using real market data clearly demonstrate a dominance relationship of our dynamic mean-multiple risk portfolio policies over the static buy-and-hold portfolio policy.

Polynomially Solvable Cases of Binary Quadratic Programs

We summarize in this chapter polynomially solvable subclasses of binary quadratic programming problems studied in the literature and report some new polynomially solvable subclasses revealed in our recent research. It is well known that the binary quadratic programming program is NP-hard in general. Identifying polynomially solvable subclasses of binary quadratic programming problems not only offers theoretical insight into the complicated nature of the problem but also provides platforms to design relaxation schemes for exact solution methods. We discuss and analyze in this chapter six polynomially solvable subclasses of binary quadratic programs, including problems with special structures in the matrix Q of the quadratic objective function, problems defined by a special graph or a logic circuit, and problems characterized by zero duality gap of the SDP relaxation. Examples and geometric illustrations are presented to provide algorithmic and intuitive insights into the problems.

Brief paper: Linear-quadratic switching control with switching cost

We study in this paper the linear-quadratic (LQ) optimal control problem of discrete-time switched systems with a constant switching cost for both finite and infinite time horizons. We reduce these problems into an auxiliary problem, which is an LQ optimal switching control problem with a cardinality constraint on the total number of switchings. Based on the solution structure derived from the dynamic programming (DP) procedure, we develop a lower bounding scheme by exploiting the monotonicity of the Riccati difference equation. Integrating such a lower bounding scheme into a branch and bound (BnB) framework, we offer an efficient numerical solution scheme for the LQ switching control problem with switching cost.

Reachability determination in acyclic Petri nets by cell enumeration approach

Reachability is one of the most important behavioral properties of Petri nets. We propose in this paper a novel approach for solving the fundamental equation in the reachability analysis of acyclic Petri nets, which has been known to be NP-complete. More specifically, by adopting a revised version of the cell enumeration method for an arrangement of hyperplanes in discrete geometry, we develop an efficient solution scheme to identify firing count vector solution(s) to the fundamental equation on a bounded integer set, with a complexity bound of O ( ( n u ) n - m ) , where n is the number of transitions, m is the number of places and u is the upper bound of the number of firings for all individual transitions.

Performance-First Control for Discrete-Time LQG Problems

Performance-first control for discrete-time LQG is considered in this paper to minimize the probability that the performance index exceeds a preselected threshold via constructing a closed-loop feedback control law. This problem can be converted into a mean-variance control problem which can be solved by developing a nested form of the variance and using polynomial optimization as a solution scheme.

Damping Characteristics of Beams with Enhanced Self-Sensing Active Constrained Layer Treatments Under Various Boundary Conditions

In this paper, an energy-based approach is developed to investigate damping characteristics of beams with enhanced self-sensing active constrained layer (ESACL) damping treatments. Analytical formulations for the active, passive, and total hybrid modal loss factors of the cantilever and simply-supported beams partially covered with the ESACL are derived. The analytical formulations are validated with the results in the literature and experimental data for the cantilever beam. Beams with other boundary conditions can also be solved and discussed using the presented approach. The results show that the edge elements in the ESACL can significantly improve the system damping performance as compared to the active constrained layer damping treatment. The effects of key parameters, such as control gain, edge element stiffness, location, and coverage of the ESACL patch on the system loss factors, have been investigated. It has also been shown that the boundary conditions play an important role on the damping characteristics of the beam structure with the ESACL treatment. With careful analysis on the location and coverage of the partially covered ESACL treatment, effective vibration control for beams under various boundary conditions for specific modes of interest would be achieved.

A polynomial case of convex integer quadratic programming problems with box integer constraints

In this paper, we study a special class of convex quadratic integer programming problem with box constraints. By using the decomposition approach, we propose a fixed parameter polynomial time algorithm for such a class of problems. Given a problem with size