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A New SelfDual Embedding Method for Convex Programming
 Journal of Global Optimization
, 2001
"... In this paper we introduce a conic optimization formulation for inequalityconstrained convex programming, and propose a selfdual embedding model for solving the resulting conic optimization problem. The primal and dual cones in this formulation are characterized by the original constraint function ..."
Abstract

Cited by 9 (2 self)
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In this paper we introduce a conic optimization formulation for inequalityconstrained convex programming, and propose a selfdual embedding model for solving the resulting conic optimization problem. The primal and dual cones in this formulation are characterized by the original constraint functions and their corresponding conjugate functions respectively. Hence they are completely symmetric. This allows for a standard primaldual path following approach for solving the embedded problem. Moreover, there are two immediate logarithmic barrier functions for the primal and dual cones. We show that these two logarithmic barrier functions are conjugate to each other. The explicit form of the conjugate functions are in fact not required to be known in the algorithm. An advantage of the new approach is that there is no need to assume an initial feasible solution to start with. To guarantee the polynomiality of the pathfollowing procedure, we may apply the selfconcordant barrier theory of Nesterov and Nemirovski. For this purpose, as one application, we prove that the barrier functions constructed this way are indeed selfconcordant when the original constraint functions are convex and quadratic. Keywords: Convex Programming, Convex Cones, SelfDual Embedding, SelfConcordant Barrier Functions. # Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong. Research supported by Hong Kong RGC Earmarked Grants CUHK4181/00E and CUHK4233/01E. 1 1
Discretetime financial planning models under lossaverse preferences
 Operations Research
, 2005
"... doi 10.1287/opre.1040.0182 ..."
Optioned Portfolio Selection: Models and Analysis ∗
, 2006
"... The meanvariance model of Markowitz and many of its extensions have been playing an instrumental role in guiding the practice of portfolio selection. In this paper we study a meanvariance formulation for the portfolio selection problem involving options. In particular, the portfolio in question con ..."
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The meanvariance model of Markowitz and many of its extensions have been playing an instrumental role in guiding the practice of portfolio selection. In this paper we study a meanvariance formulation for the portfolio selection problem involving options. In particular, the portfolio in question contains a stock index and some European style options on the index. A refined meanvariance methodology is adopted in our approach to formulate this problem as multistage stochastic optimization. It turns out that there are two different solution techniques, both lead to explicit solutions of the problem: one is based on stochastic programming and optimality conditions, and the other one is based on stochastic control and dynamic programming. We introduce both techniques, because their strengths are very different so as to suit different possible extensions and refinements of the basic model. Attention is paid to the structure of the optimal payoff function, which is shown to possess rich properties. Further refinements of the model, such as the request that the payoff should be monotonic with respect to the index, are discussed. Throughout the paper, various numerical examples are used to illustrate the underlying concepts.
An InteriorPoint and Decomposition Approach to Multiple Stage Stochastic Programming
"... ounded by the demand #. Finally, z # is the amount of newspapers to be returned to the publisher at the end of the day. A closer look suggests that the objective is actually to maximize the expected profit, which is the revenue minus the costs. The model looks like a pretty ordinary optimization pr ..."
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ounded by the demand #. Finally, z # is the amount of newspapers to be returned to the publisher at the end of the day. A closer look suggests that the objective is actually to maximize the expected profit, which is the revenue minus the costs. The model looks like a pretty ordinary optimization problem, except that there is a random variable # in the constraint. The model is known as twostage stochjj linear programming, as it involves decisions at two stages: (1) the decision at the beginning of the day, namely x; (2) the decision to be made during the day, namely y # and z # , given that x is already decided. Moreover, all the relationships happen to be linear. Clearly, one may consider extended models where more than two stages of decisions have to be made sequentially, depending on the newly arrived information on the uncertain factors. That more general case is naturally termed multistage stochjjB linear programming. Since # has only two possibly outcomes in our case, we may wri
Multistage Stochastic Linear Programming: An Approach by Events
"... To solve the multistage linear programming problem, one may use a deterministic or a stochastic approach. The drawbacks of the two techniques are well known: the deterministic approach is unrealistic under uncertainty and the stochastic approach suffers from scenario explosion. We introduce a new t ..."
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To solve the multistage linear programming problem, one may use a deterministic or a stochastic approach. The drawbacks of the two techniques are well known: the deterministic approach is unrealistic under uncertainty and the stochastic approach suffers from scenario explosion. We introduce a new technique, whose objective is to overcome both drawbacks. The focus of this new technique is on events instead of scenarios and for this reason we call it Multistage Event Linear Programming (MELP). As we show in the theoretical results and in the preliminary computational experiments, the MELP approach represents a promising compromise between the stochastic and the deterministic approach, regarding capacity to deal with uncertainty and computational tractability.
Multistage Stochastic Linear Programming: Scenarios Versus Events
"... To solve the multistage linear programming problem, one may use a deterministic or a stochastic approach. The drawbacks of the two techniques are well known: the deterministic approach is unrealistic under uncertainty and the stochastic approach suffers from scenario explosion. We introduce a new s ..."
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To solve the multistage linear programming problem, one may use a deterministic or a stochastic approach. The drawbacks of the two techniques are well known: the deterministic approach is unrealistic under uncertainty and the stochastic approach suffers from scenario explosion. We introduce a new scheme, whose objective is to overcome both drawbacks. The focus of this new scheme is on events instead of scenarios and for this reason we call it Multistage Event Linear Programming (ELP). As we show in the theoretical results and in the preliminary computational experiments, the ELP approach represents a promising compromise between the stochastic and the deterministic approach, regarding capacity to deal with uncertainty and computational tractability.