Parametric Semidefinite Programming: Geometry of the Trajectory of Solutions

TL;DR

It is shown that only six distinct behaviors can be observed at a neighborhood of a given point along the solution trajectory, defined as the set of solutions depending on a time parameter.

Abstract

In many applications, solutions of convex optimization problems are updated on-line, as functions of time. In this paper, we consider parametric semidefinite programs, which are linear optimization problems in the semidefinite cone whose coefficients (input data) depend on a time parameter. We are interested in the geometry of the solution (output data) trajectory, defined as the set of solutions depending on the parameter. We propose an exhaustive description of the geometry of the solution trajectory. As our main result, we show that only six distinct behaviors can be observed at a neighborhood of a given point along the solution trajectory. Each possible behavior is then illustrated by an example.

Figures (5)

• Figure 1: Trajectory of solutions of (P$^1_t)$. Its feasible set is time-invariant and it is the Cayley spectrahedron (orange). Its optimal set-valued map coincides with the red dot at $(1,1,1)$ for $t\in(-3,-2]$, moves along the blue curve $(-t/2,-t/2,t^2/2-1 )$ for $t\in(-2,2)\setminus\{0\}$, and covers the whole red top edge $\{(x,y,-1)|x+y=0\}$ at $t=0$.
• Figure 2: Trajectory of solutions of (P$^2_t)$. Its feasible set is time-invariant and it is the Cayley spectrahedron (orange) intersected with half space $\{(x,y,z)|1+x+y+z\ge0\}$ (green). Its optimal set-valued map moves along the blue curve $(-t/2,-t/2,t^2/2-1 )$ for $t\in(-1,0)$, and covers the whole red top edge $\{(x,y,z)|x+y=0,z=-1\}$ for $t\in[0,1)$.
• Figure 3: Trajectory of solutions of (D$^3_t)$. Its feasible set is time-invariant and it is the set $\{(x,y,z)|y+z^2\le0,\; -z\le x\le z\}$ (orange). Its optimal set-valued map coincides with the red dot at $(0,0,0)$ for $t\in(-1,0]$. At $t=0$, $(0,0,0)$ is a continuous bifurcation point, as for every $t\in(0,1)$ the solution is multi-valued and equal to the set $\{(x,y,z)|x\in[-t/2,t/2], y=-t^2/4, z=-t/2\}$. In the picture, the blue segments illustrate the optimal multiple-valued solution for $t=\{0.1,0.2,\dots,0.9,1\}$
• Figure 4: Graph of the $x$ coordinate of the optimal set of $(P^4_t)$ as a function of time $t$. The blue segments correspond to regular points, the red dot corresponds to an irregular accumulation point, and the orange vertical segments correspond to discontinuous isolated multiple-points, where the solution is multiple valued.
• Figure 5: Graph of the $x$ coordinate of the optimal set of (P$^5_t)$ as a function of time $t$. The blue segment consists of regular points, the red dot corresponds to an irregular accumulation point, and the orange dots correspond to continuous bifurcation points. The gray region corresponds to times intervals where the optimal solution is multi-valued.

Theorems & Definitions (39)

• Theorem 1.1: Informal statement of Theorem \ref{['thm: main_result']}
• Theorem 1.2: Informal statement of Theorem \ref{['thm: finite_bad_result']}
• Remark 2.1
• Definition 2.2: KKT conditions
• Definition 2.3: Strict feasibility
• Definition 2.4: Strict complementarity
• Definition 2.5: Primal non-degeneracy
• Definition 2.6: Dual non-degeneracy
• Definition 2.7: Non-degeneracy
• Proposition 2.8