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Register a new userSecond-order cone optimization of the gradostat
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Josh Taylor
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We maximize the production of biogas in a gradostat at steady state. The physical decision variables are the water, substrate, and biomass entering each tank and the flows through the interconnecting pipes. Our main technical focus is the nonconvex constraint describing microbial growth. We formulate a relaxation and prove that it is exact when the gradostat is outflow connected, its system matrix is irreducible, and the growth rate satisfies a simple condition. The relaxation has second-order cone representations for the Monod and Contois growth rates. We extend the steady state models to the case of multiple time periods by replacing the derivatives with numerical approximations instead of setting them to zero. The resulting optimizations are second-order cone programs, which can be solved at large scales using standard industrial software.
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This paper is devoted to the study of tilt stability of local minimizers, which plays an important role in both theoretical and numerical aspects of optimization. This notion has been comprehensively investigated in the unconstrained framework as well as for problems of nonlinear programming with $C^2$-smooth data. Available results for nonpolyhedral conic programs were obtained only under strong constraint nondegeneracy assumptions. Here we develop an approach of second-order variational analysis, which allows us to establish complete neighborhood and pointbased characterizations of tilt stability for problems of second-order cone programming generated by the nonpolyhedral second-order/Lorentz/ice-cream cone. These characterizations are established under the weakest metric subregularity constraint qualification condition.
The error bound property for a solution set defined by a set-valued mapping refers to an inequality that bounds the distance between vectors closed to a solution of the given set by a residual function. The error bound property is a Lipschitz-like/calmness property of the perturbed solution mapping, or equivalently the metric subregularity of the underlining set-valued mapping. It has been proved to be extremely useful in analyzing the convergence of many algorithms for solving optimization problems, as well as serving as a constraint qualification for optimality conditions. In this paper, we study the error bound property for the solution set of a very general second-order cone complementarity problem (SOCCP). We derive some sufficient conditions for error bounds for SOCCP which is verifiable based on the initial problem data.
We consider linear optimization over a fixed compact convex feasible region that is semi-algebraic (or, more generally, tame). Generically, we prove that the optimal solution is unique and lies on a unique manifold, around which the feasible region is partly smooth, ensuring finite identification of the manifold by many optimization algorithms. Furthermore, second-order optimality conditions hold, guaranteeing smooth behavior of the optimal solution under small perturbations to the objective.
We introduce a conic embedding condition that gives a hierarchy of cones and cone programs. This condition is satisfied by a large number of convex cones including the cone of copositive matrices, the cone of completely positive matrices, and all symmetric cones. We discuss properties of the intermediate cones and conic programs in the hierarchy. In particular, we demonstrate how this embedding condition gives rise to a family of cone programs that interpolates between LP, SOCP, and SDP. This family of $k$th order cones may be realized either as cones of $n$-by-$n$ symmetric matrices or as cones of $n$-variate even degree polynomials. The cases $k = 1, 2, n$ then correspond to LP, SOCP, SDP; or, in the language of polynomial optimization, to DSOS, SDSOS, SOS.
In this paper we study second-order optimality conditions for non-convex set-constrained optimization problems. For a convex set-constrained optimization problem, it is well-known that second-order optimality conditions involve the support function of the second-order tangent set. In this paper we propose two approaches for establishing second-order optimality conditions for the non-convex case. In the first approach we extend the concept of the support function so that it is applicable to general non-convex set-constrained problems, whereas in the second approach we introduce the notion of the directional regular tangent cone and apply classical results of convex duality theory. Besides the second-order optimality conditions, the novelty of our approach lies in the systematic introduction and use, respectively, of direction
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