Abstract
In this work we investigate theportfolio selection problem(P1) andbi-directional trading(P2) when prices are interrelated. Zhanget al.(J. Comb. Optim.23(2012) 159–166) provided the algorithm UND which solves one variant of P2. We are interested in solutions which are optimal from a worst-case perspective. For P1, we prove the worst-case input sequence and derive the algorithmoptimal portfolio for interrelated prices(OPIP). We then prove the competitive ratio and optimality. We use the idea of OPIP to solve P2 and derive the algorithm calledoptimal conversion for interrelated prices(OCIP). Using OCIP, we also design optimal online algorithms forbi-directional search(P3) calledbi-directional UND(BUND) andoptimal online search for unknown relative price bounds(RUN). We run numerical experiments and conclude that OPIP and OCIP perform well compared to other algorithms even if prices do not behave adverse.
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