On the weak second-order optimality condition for nonlinear semidefinite and second-order cone programming

Second-order necessary optimality conditions for nonlinear conic programming problems that depend on a single Lagrange multiplier are usually built under nondegeneracy and strict complementarity. In this paper we establish a condition of such type for two classes of nonlinear conic problems, namely semidefinite and second-order cone programming, assuming Robinson’s constraint qualification and a generalized form … Read more

Modular-topology optimization with Wang tilings: An application to truss structures

Modularity is appealing for solving many problems in optimization. It brings the benefits of manufacturability and reconfigurability to structural optimization, and enables a trade-off between the computational performance of a Periodic Unit Cell (PUC) and the efficacy of non-uniform designs in multi-scale material optimization. Here, we introduce a novel strategy for concurrent minimum-compliance design of … Read more

Rational Polyhedral Outer-Approximations of the Second-Order Cone

It is well-known that the second-order cone can be outer-approximated to an arbitrary accuracy by a polyhedral cone of compact size defined by irrational data. In this paper, we propose two rational polyhedral outer-approximations of compact size retaining the same guaranteed accuracy. The first outer-approximation has the same size as the optimal but irrational outer-approximation … Read more

Exploiting Aggregate Sparsity in Second Order Cone Relaxations for Quadratic Constrained Quadratic Programming Problems

Among many approaches to increase the computational efficiency of semidefinite programming (SDP) relaxation for quadratic constrained quadratic programming problems (QCQPs), exploiting the aggregate sparsity of the data matrices in the SDP by Fukuda et al. (2001) and second-order cone programming (SOCP) relaxation have been popular. In this paper, we exploit the aggregate sparsity of SOCP … Read more

Detection and Transformation of Second-Order Cone Programming Problems in a General-Purpose Algebraic Modeling Language

Diverse forms of nonlinear optimization problems can be recast to the special form of second-order cone problems (SOCPs), permitting a wider variety of highly effective solvers to be applied. Popular solvers assume, however, that the necessary transformations to required canonical forms have already been identified and carried out. We describe a general approach to the … Read more

Robust Sensitivity Analysis for Linear Programming with Ellipsoidal Perturbation

Given an originally robust optimal decision and allowing perturbation parameters of the linear programming problem to run through a maximum uncertainty set controlled by a variable of perturbation radius, we do robust sensitivity analysis for the robust linear programming problem in two scenarios. One is to keep the original decision still robust optimal, the other … Read more

DSOS and SDSOS Optimization: More Tractable Alternatives to Sum of Squares and Semidefinite Optimization

In recent years, optimization theory has been greatly impacted by the advent of sum of squares (SOS) optimization. The reliance of this technique on large-scale semidefinite programs however, has limited the scale of problems to which it can be applied. In this paper, we introduce DSOS and SDSOS optimization as linear programming and second-order cone … Read more

A primal-dual interior-point method based on various selections of displacement step for second-order cone programming

In this paper, a primal-dual interior-point method equipped with various selections of the displacement step are derived for solving second-order cone programming problems. We first establish the existence and uniqueness of the optimal solution of the corresponding perturbed problem and then demonstrate its convergence to the optimal solution of the original problem. Next, we present … Read more

A logarithmic barrier interior-point method based on majorant functions for second-order cone programming

We present a logarithmic barrier interior-point method for solving a second-order cone programming problem. Newton’s method is used to compute the descent direction, and majorant functions are used as an efficient alternative to line search methods to determine the displacement step along the direction. The efficiency of our method is shown by presenting numerical experiments. … Read more

A New Voltage Stability-Constrained Optimal Power Flow Model: Sufficient Condition, SOCP Representation, and Relaxation

A simple characterization of the solvability of power flow equations is of great importance in the monitoring, control, and protection of power systems. In this paper, we introduce a sufficient condition for power flow Jacobian nonsingularity. We show that this condition is second-order conic representable when load powers are fixed. Through the incorporation of the … Read more