Linear Programming with Inequality Constraints via Entropic Perturbation

A dual convex programming approach to solving linear programs with inequality constraints through entropic perturbation is derived. The amount of perturbation required depends on the desired accuracy of the optimum. The dual program contains only non-positivity constraints. An ϵ-optimal solution to the linear program can be obtained effortlessly from the optimal solution of the dual program. Since cross-entropy minimization subject to linear inequality constraints is a special case of the perturbed linear program, the duality result becomes readily applicable. Many standard constrained optimization techniques can be specialized to solve the dual program. Such specializations, made possible by the simplicity of the constraints, significantly reduce the computational effort usually incurred by these methods. Immediate applications of the theory developed include an entropic path-following approach to solving linear semi-infinite programs with an infinite number of inequality constraints and the widely used entropy optimization models with linear inequality and/or equality constraints

Entropic Regularization Approach for Mathematical Programs with Equilibrium Constraints

A new smoothing approach based on entropic perturbationis proposed for solving mathematical programs withequilibrium constraints. Some of the desirableproperties of the smoothing function are shown. Theviability of the proposed approach is supported by acomputationalstudy on a set of well-known test problems.mathematical programs with equilibrium constraints;entropic regularization;smoothing approach

Entropic regularization approach for mathematical programs with equilibrium constraints

A new smoothing approach based on entropic perturbation is proposed for solving mathematical programs with equilibrium constraints. Some of the desirable properties of the smoothing function are shown. The viability of the proposed approach is supported by a computational study on a set of well-known test problems.Entropic regularization;Smoothing approach;Mathematical programs with equilibrium constraints

Entropy flow in near-critical quantum circuits

Near-critical quantum circuits are ideal physical systems for asymptotically large-scale quantum computers, because their low energy collective excitations evolve reversibly, effectively isolated from the environment. The design of reversible computers is constrained by the laws governing entropy flow within the computer. In near-critical quantum circuits, entropy flows as a locally conserved quantum current, obeying circuit laws analogous to the electric circuit laws. The quantum entropy current is just the energy current divided by the temperature. A quantum circuit made from a near-critical system (of conventional type) is described by a relativistic 1+1 dimensional relativistic quantum field theory on the circuit. The universal properties of the energy-momentum tensor constrain the entropy flow characteristics of the circuit components: the entropic conductivity of the quantum wires and the entropic admittance of the quantum circuit junctions. For example, near-critical quantum wires are always resistanceless inductors for entropy. A universal formula is derived for the entropic conductivity: \sigma_S(\omega)=iv^{2}S/\omega T, where \omega is the frequency, T the temperature, S the equilibrium entropy density and v the velocity of `light'. The thermal conductivity is Real(T\sigma_S(\omega))=\pi v^{2}S\delta(\omega). The thermal Drude weight is, universally, v^{2}S. This gives a way to measure the entropy density directly.Comment: 2005 paper published 2017 in Kadanoff memorial issue of J Stat Phys with revisions for clarity following referee's suggestions, arguments and results unchanged, cross-posting now to quant-ph, 27 page

A recursive algorithm for multivariate risk measures and a set-valued Bellman's principle

A method for calculating multi-portfolio time consistent multivariate risk measures in discrete time is presented. Market models for $d$ assets with transaction costs or illiquidity and possible trading constraints are considered on a finite probability space. The set of capital requirements at each time and state is calculated recursively backwards in time along the event tree. We motivate why the proposed procedure can be seen as a set-valued Bellman's principle, that might be of independent interest within the growing field of set optimization. We give conditions under which the backwards calculation of the sets reduces to solving a sequence of linear, respectively convex vector optimization problems. Numerical examples are given and include superhedging under illiquidity, the set-valued entropic risk measure, and the multi-portfolio time consistent version of the relaxed worst case risk measure and of the set-valued average value at risk.Comment: 25 pages, 5 figure