A strict complementarity approach to error bound and sensitivity of solution of conic programs

Abstract

In this paper, we provide an elementary, geometric, and unified framework to analyze conic programs that we call the strict complementarity approach. This framework allows us to establish error bounds and quantify the sensitivity of the solution. The framework uses three classical ideas from convex geometry and linear algebra: linear regularity of convex sets, facial reduction, and orthogonal decomposition. We show how to use this framework to derive error bounds for linear programming, second order cone programming, and semidefinite programming.

Appendices

Appendix A: Proof for Section 4

We first show the Slater’s condition for ($\mathcal {P}'$) and its dual. Recall the Slater’s condition for the original ($\mathcal {P}$) means that the two points $x_0,y_0$ satisfying $(x_0,c-\mathcal {A}^*y)\in \mathbf{int}(\mathcal {K}) \times \mathbf{int}(\mathcal {K}^*)$. This implies that $\min \{\mathbf{dist{}}(x_0,\partial \mathcal {K}),\mathbf{dist{}}(c-\mathcal {A}^*y_0,\partial \mathcal {K}^*)\}\ge \eta$ for some constant $\eta >0$. We now construct $x_0'$ and $y_0'$ from $x_0$ and $y_0$. It can be easily verified if $\left\| \Delta \mathcal {A}\right\| \le \frac{\sigma _{\min >0}(\mathcal {A})}{2}$, $\frac{2}{\sigma _{\min >0}(\mathcal {A})}(\left\| \Delta b\right\| +\left\| \Delta \mathcal {A}\right\| \left\| x_0\right\| )\le \frac{\eta }{2}$, and $\left\| \Delta c\right\| +\left\| \Delta \mathcal {A}\right\| \left\| y_0\right\| \le \frac{\eta }{2}$, then the choice

$$\begin{aligned} \begin{aligned} x_0' = x_0 +\mathcal {A'}^\dagger (\Delta b-\Delta \mathcal {A}x_0), \quad y_0' = y_0 \end{aligned} \end{aligned}$$

(16)

satisfy $\min \{\mathbf{dist{}}(x_0',\partial \mathcal {K}),\mathbf{dist{}}(c'-(\mathcal {A}^*)'y_0', \partial (\mathcal {K}^*))\}\ge \frac{\eta }{2}$, $\max \{ \left\| x_0-x_0'\right\| ,\left\| y_0-y_0'\right\| \le \frac{\eta }{2}$, and are feasible for ($\mathcal {P}'$) and its dual.

Now for the boundedness condition of any solution $x_\star '$ to ($\mathcal {P}'$). Using the previous constructed $x_0'$ and $y_0'$, we know

$$\begin{aligned} \begin{aligned} \langle x_\star ', c'-(\mathcal {A}^*)'y_0' \rangle&\le \langle x_0', c'-(\mathcal {A}^*)'y_0' \rangle =\langle c', x_0' \rangle -\langle b', y_0' \rangle \\&=\langle c, x_0 \rangle -\langle b, y_0 \rangle + \langle \Delta c, x_0' \rangle -\langle \Delta b, y_0' \rangle \\&\le \langle c, x_0 \rangle -\langle b, y_0 \rangle +\frac{\eta }{2}(\left\| x_0\right\| +\frac{\eta }{2}+ \frac{\sigma _{\min >0}(\mathcal {A})}{2}\left\| y_0\right\| ) \end{aligned} \end{aligned}$$

(17)

The rest is a simple consequence of the following lemma.

Lemma 5

Suppose $\mathcal {K}$ is closed and convex. Given $\epsilon >0$ and $s_0\in (\mathcal {K}^*)^\circ$ with $d = \mathbf{dist{}}(s_0,\partial \mathcal {K}^*)>0$, there is a $B>0$ such that for any x satisfying $x\in \mathcal {K}$, and $\langle x, s \rangle \le \epsilon$ for some s with $\left\| s-s_0\right\| \le \frac{d}{2}$, its norm satisfies $\left\| x\right\| \le B$.

Proof

Suppose such B does not exist, then there is a sequence $(x_n,s_n)\in \mathcal {K}\times \mathcal {K}^*$ with $\left\| s_n-s_0\right\| \le \frac{d}{2}$, $\langle x_n, s_n \rangle \le \epsilon$, and $\lim _{n\rightarrow \infty }\left\| x_n\right\| =+\infty$.

Now consider $(\frac{x_n}{\left\| x_n\right\| },s_n)\in \mathcal {K}\times \mathcal {K}^*$. Since $\frac{x_n}{\left\| x_n\right\| }$ and $s_n$ are bounded, we can choose a appropriate subsequence of $(\frac{x_n}{\left\| x_n\right\| },s_n)$ which converges to certain $(x,s)\in \mathcal {K}\times \mathbf{int}(\mathcal {K}^*)$ as $\left\| s_n-s_0\right\| \le \frac{d}{2}$. Call the subsequence $(\frac{x_n}{\left\| x_n\right\| },s_n)$ still. Using $\langle \frac{x_n}{\left\| x_n\right\| }, s_n \rangle \le \frac{\epsilon }{\left\| x_n\right\| }$ and $\left\| x_n\right\| \rightarrow +\infty$, we see

$$\begin{aligned} \langle x, s \rangle =0. \end{aligned}$$

This is not possible as $s\in \mathbf{int}(\mathcal {K}^*)$. Hence such B must exist.

Appendix B: Equivalence between DSC and SC for LP and SDP

We define the strict complementarity of LP and SDP, and show it is equivalent to DSC defined in Sect. 2.1. For a vector $x\in { \mathbf{R}}^{n}$, denote ${\mathbf {nnz}}(x)$ as its number of nonzeros.

Definition 3

For LP, if there exists optimal primal dual pair $(x_\star ,y_\star )\in \mathcal {X}_\star \times \mathcal {Y}_\star \subset { \mathbf{R}}_+^n\times { \mathbf{R}}^m$ with $s_\star = c-\mathcal {A}^* y_\star \in { \mathbf{R}}_+^n$ such that

$$\begin{aligned} {\mathbf {nnz}}(x_\star )+{\mathbf {nnz}}(s_\star )=n, \end{aligned}$$

we say ($\mathcal {P}$) satisfies strict complementarity. Similary, for SDP, if there exists optimal primal dual pair $(X_\star ,y_\star )\in \mathcal {X}_\star \times \mathcal {Y}_\star \subset { \mathbf{S}}_+^n\times { \mathbf{R}}^m$ with $S_\star = C-\mathcal {A}^* y_\star \in { \mathbf{S}}_+^n$ such that

$$\begin{aligned} {\mathbf{rank}}(X_\star )+{\mathbf{rank}}(S_\star )=n, \end{aligned}$$

we say ($\mathcal {P}$) satisfies strict complementarity.

Lemma 6

For both LP and SDP, under strong duality, the strict complementarity defined is equivalent to dual strict complementarity.

Proof

For LP, under strong duality (see (SD)), we have for any optimal $x_\star$, $x_\star \in \mathcal {F}_{s_\star }=\{x \in { \mathbf{R}}_+^n\mid x_i = 0,\text { for all } (s_\star )_i>0\}$. The relative interior of $\mathcal {F}_{s_\star }$ is

$$\begin{aligned} \mathrm{rel}(\mathcal {F}_{s_\star })= \{x \in { \mathbf{R}}_+^n\mid x_i = 0\;\text { for all } (s_\star )_i>0,\; \text {and}\; x_i >0 \;\text { for all } (s_\star )_i=0\}. \end{aligned}$$

The equivalence between DSC and SC is then immediate.

For SDP, under strong duality (see (SD)), we know that for any optimal $X_\star$, $X_\star \in \mathcal {F}_{S_\star } = \{X\mid \langle X, S_\star \rangle =0, X\succeq 0\}=\{X\mid {\mathbf{range}}(X)\subset { \mathbf{nullspace}}(S_\star ),\; \text {and}\; X\succeq 0\}$. The relative interior of $\mathcal {F}_{S_\star }$ is

$$\begin{aligned} \mathrm{rel}(\mathcal {F}_{S_\star }) = \{X\mid {\mathbf{range}}(X)={ \mathbf{nullspace}}(S_\star ),\; \text {and}\; X\succeq 0\}. \end{aligned}$$

The equivalence is immediate by using Rank-nullity theorem for $S_\star$.

Appendix C: A lower bound on distance to optimality: $\left\| \mathcal {P}_{\mathcal {V}_{s_\star }^\perp }(x)\right\| _2 \le \mathbf{dist{}}(x,\mathcal {X}_\star )$

We have shown how to establish upper bounds on $\mathbf{dist{}}(x,\mathcal {X}_\star )$ with an upper bound on $\left\| \mathcal {P}_{\mathcal {V}{s_\star }^\perp }(x_{+})\right\| _2$. In this section, we show that the same quantity $\left\| \mathcal {P}_{\mathcal {V}{s_\star }^\perp }(x_{+})\right\| _2$ also yields a lower bound. Hence, it is important to understand the behavior of $\left\| \mathcal {P}_{\mathcal {V}{s_\star }^\perp }(x_{+})\right\| _2$. For simplicity, we suppose in this section that x is feasible for the problem ($\mathcal {P}$) : in this case, $\left\| \mathcal {P}_{\mathcal {V}{s_\star }^\perp }(x_{+})\right\| _2 = \left\| \mathcal {P}_{\mathcal {V}{s_\star }^\perp }(x)\right\| _2$. The argument for infeasible x is essentially the same.

First recall for feasible x, the only nonzero error metric is the suboptimality $\epsilon _{{\tiny opt}}(x)=\langle c, x \rangle -p_\star$. Also note that $\epsilon _{{\tiny opt}}(x)=0 \iff \left\| \mathcal {P}_{\mathcal {V}_{s_\star }^\perp }(x)\right\| _2$ using complementary slackness. Hence, there is some nonnegative function $g:{ \mathbf{R}}\rightarrow { \mathbf{R}}_+$ such that $g(\epsilon _{{\tiny opt}}(x))\le \left\| \mathcal {P}_{\mathcal {V}_{s_\star }^\perp }(x)\right\| _2$. Now note that for any feasible x, we always have the lower bound on distance given by $\left\| \mathcal {P}_{\mathcal {V}_{s_\star }^\perp }(x)\right\| _2$ as $\mathcal {V}_{s_\star }\supset \mathcal {X}_\star$,

$$\begin{aligned} g(\epsilon _{{\tiny opt}}(x))\le \left\| \mathcal {P}_{\mathcal {V}_{s_\star }^\perp }(x)\right\| _2 = \mathbf{dist{}}(x,\mathcal {V}_{s_\star })\le \mathbf{dist{}}(x,\mathcal {X}_\star ). \end{aligned}$$

(18)

The lower bound (18) hence shows that $\left\| \mathcal {P}_{\mathcal {V}_{s_\star }^\perp }(x_{+})\right\| _2$ provides a lower bound on the distance $\mathbf{dist{}}(x,\mathcal {X}_\star )$, and provides hope that this bound might scale with the suboptimality $\epsilon _{{\tiny opt}}(x)$. We summarize our findings in the following theorem.

Theorem 3

Suppose there exists a increasing continuous g with $g(0)=0$ so that for any x feasible for Problem ($\mathcal {P}$)⁸ with $\epsilon (x)\le \bar{c}$,

$$\begin{aligned} \begin{aligned} \left\| \mathcal {P}_{\mathcal {V}_{s_\star }^\perp }(x)\right\| _2 \ge g(\epsilon _{{\tiny opt}}(x)). \end{aligned} \end{aligned}$$

(19)

Then the following inequality holds:

$$\begin{aligned} \begin{aligned} \mathbf{dist{}}(x,\mathcal {X}_\star ) \ge g(\epsilon _{{\tiny opt}}(x)). \end{aligned} \end{aligned}$$

(20)

Remark 5

We also have a partial converse for the above theorem that follows from the same proof. Assume $\mathbf{dist{}}(\mathcal {P}_{\mathcal {V}_{s_\star }}(x),\mathcal {F}_{s_\star })$ is 0. If the relation (20) holds for all feasible x with $\epsilon _{{\tiny opt}}(x)\le \bar{c}$, then $\left\| \mathcal {P}_{\mathcal {V}_{s_\star }^\perp }(x)\right\| _2 \ge \frac{g(\epsilon _{{\tiny opt}}(x))}{1+\gamma \sigma _{\max }(\mathcal {A})}$. Here $\gamma$ is defined in (3).

Proof

We have proved that (19) implies (20) using the motivating logic laid out at the beginning of this section.

Conversely, assume the term $\mathbf{dist{}}(\mathcal {P}_{\mathcal {V}_{s_\star }}(x_{+}),\mathcal {F}_{s_\star })$ is zero, x is feasible, $\epsilon _{{\tiny opt}}(x)\le \bar{c}$, and the inequality (20) holds. We see that for all feasible x,

$$\begin{aligned} \begin{aligned} \beta g(\epsilon _{{\tiny opt}}(x)) \le \mathbf{dist{}}(x,\mathcal {X}_\star ) \overset{(a)}{\le } (1+\gamma \sigma _{\max }(\mathcal {A}))\left\| \mathcal {P}_{\mathcal {V}_{s_\star }^\perp }(x)\right\| _2. \end{aligned} \end{aligned}$$

(21)

where we use (7) for step (a).

Appendix D: Conic decomposition

In the main paper, we decompose a general $x\in \mathbf {E}$ according to the subspace $\mathcal {V}_{s_\star }$. A different decomposition uses the cone $\mathcal {F}_{s_\star }$: every $x\in \mathbf {E}$ admits the conic decomposition $x = \mathcal {P}_{\mathcal {F}_{s_\star }}(x) + \mathcal {P}_{\mathcal {F}_{s_\star }^{\circ }}(x)$ where $\mathcal {F}_{s_\star }^{\circ }$ is the polar cone of $\mathcal {F}_{s_\star }$, i.e., the negative dual cone $-\mathcal {F}^{*}_{s_\star }$.

Theorem 4

Suppose strong duality and dual strict complementarity hold. Then for some constants $\gamma ,\gamma '$ described in Lemma 2 and for all $x\in \mathbf {E}$, we have

$$\begin{aligned} \begin{aligned} \mathbf{dist{}}(x,\mathcal {X}_\star ) \le&(1+\gamma \sigma _{\max }(\mathcal {A}))\left\| \mathcal {P}_{\mathcal {F}_{s_\star }^\circ }(x)\right\| _2+\gamma \left\| \mathcal {A}(x)-b\right\| _2. \end{aligned} \end{aligned}$$

(22)

Let us compare the above bound (22) and (4) in Theorem 1. To make the comparison easier, first note that from the proof of Theorem 1, we can bound the distance from x to $\mathcal {X}_\star$ using the decomposition $x = \mathcal {P}_{\mathcal {V}_{s_\star }}(x) +\mathcal {P}_{\mathcal {V}_{s_\star }^\perp }(x)$:

$$\begin{aligned} \begin{aligned} \mathbf{dist{}}(x,\mathcal {X}_\star ) \le (1+\gamma \sigma _{\max }(\mathcal {A}))\left\| \mathcal {P}_{\mathcal {V}_{s_\star }^\perp }(x)\right\| _2+\gamma \left\| \mathcal {A}(x)-b\right\| _2 + \gamma '\mathbf{dist{}}(\mathcal {P}_{\mathcal {V}_{s_\star }}(x),\mathcal {F}_{s_\star }). \end{aligned} \end{aligned}$$

(23)

The bound (4) in Theorem 1 is further obtained via the decomposition $x=x_{+}+x_{-}$.

Comparing (23) and (22), we find that there is an extra term $\gamma '\mathbf{dist{}}(\mathcal {P}_{\mathcal {V}_{s_\star }}(x),\mathcal {F}_{s_\star })$ in (23) and the term $\left\| \mathcal {P}_{\mathcal {V}_{s_\star }^\perp }(x)\right\| _2$ in (23) is replaced by $\left\| \mathcal {P}_{\mathcal {F}_{s_\star }^\circ }(x)\right\| _2$. Since $\left\| \mathcal {P}_{\mathcal {F}_{s_\star }^\circ }(x)\right\| _2 \ge \left\| \mathcal {P}_{\mathcal {V}_{s_\star }^\perp }(x)\right\| _2$, it is not immediately clear which bound is tighter.

A more subtle difference between (22) and (4) in Theorem 1 is that we are not able to further bound $\left\| \mathcal {P}_{\mathcal {F}_{s_\star }^\circ }(x)\right\| _2$ using the decomposition $x=x_{+}+x_{-}$ with respect to $\mathcal K$. We reach this impasse because the projection operator $\mathcal {P}_{\mathcal {F}_{s_\star }^\circ }$ is not linear and so we cannot rely on the triangle inequality $\left\| \mathcal {P}_{\mathcal {F}_{s_\star }^\circ }(x)\right\| _2 \le \left\| \mathcal {P}_{\mathcal {F}_{s_\star }^\circ }(x_{+})\right\| _2 + \left\| \mathcal {P}_{\mathcal {F}_{s_\star }^\circ }(x_{-})\right\| _2$. Hence we cannot bound $\left\| \mathcal {P}_{\mathcal {F}_{s_\star }^\circ }(x)\right\| _2$ using conic infeasibility.

Thus, to use (23), we must use x (which may be infeasible with respect to the cone $\mathcal K$) directly to bound $\left\| \mathcal {P}_{\mathcal {F}_{s_\star }^\circ }(x)\right\| _2$. We now consider how to bound this term for the cases considered in the main text.

Case $s_\star =0$. For $s_\star =0$, we have $\mathcal {F}_{s_\star }= \mathcal {K}$. Thus $\mathcal {F}_{s_\star }^\circ = \mathcal {K}^\circ$ and $\mathcal {P}_{\mathcal {F}_{s_\star }^\circ }(x) = x_-$. For $s_\star \in \mathbf{int}(\mathcal {K}^*)$, we have $\mathcal {F}_{s_\star }=\{0\}$. Thus $\mathcal {F}_{s_\star }^\circ = \mathbf {E}$ and $\left\| \mathcal {P}_{\mathcal {F}_{s_\star }^\circ }(x)\right\| _2 =\left\| x\right\| _2\le \left\| x_+\right\| _2 + \left\| x_-\right\| _2$. We can bound $\left\| x_+\right\| _2$ as in Lemma 3.

Case $s_\star ={ \mathbf{R}}_+$. For $\mathcal {K}= { \mathbf{R}}_+$, the projection of $\mathcal {P}_{\mathcal {F}_{s_\star }^\circ }(x) = x - (x_{I_{s_\star }^c})_+= (x_+)_{I_{s_\star }} + x_-$ where $x_{I_{s_\star }^c}$ is the vector x zeroing all entries in the support of $s_\star$ and $(x_+)_{I_{s_\star }}$ is the vector $x_+$ zeroing out all entries not in the support of $s_\star$. Hence, $\left\| \mathcal {P}_{\mathcal {F}_{s_\star }^\circ }(x) \right\| _2\le \left\| (x_+)_{I_{s_\star }}\right\| _2 + \left\| x_-\right\| _2$ and we can further bound $\left\| (x_+)_{I_{s_\star }}\right\| _2$ as in Lemma 3.

Case $s_\star =\text{ SOC}_{n}$. For $\mathcal {K}= \text{ SOC}_{n}$, considering the nontrivial case $s_\star \not = 0$, the projection of $\mathcal {P}_{\mathcal {F}_{s_\star }^\circ }(x)$ is

$$\begin{aligned} x -(\langle x, \check{s}_{\star } \rangle )_+ \check{s}_{\star }= x_+ -(\langle x, \check{s}_{\star } \rangle )_+ \check{s}_{\star }+x_-. \end{aligned}$$

Thus we have

$$\begin{aligned} \begin{aligned} \left\| \mathcal {P}_{\mathcal {F}_{s_\star }^\circ }(x) - \mathcal {P}_{\mathcal {V}^\perp _{s_\star }}(x_+)\right\| _2&= \left\| \left( \langle x_+, \check{s}_{\star } \rangle - (\langle x, \check{s}_{\star } \rangle )_+ \right) \check{s}_{\star }+x_-\right\| _2 \\&\le |\langle x_+, \check{s}_{\star } \rangle - (\langle x, \check{s}_{\star } \rangle )_+ | + \left\| x_-\right\| _2 \\&\overset{(a)}{\le } |\langle x_+, \check{s}_{\star } \rangle | + \left\| x_-\right\| _2 . \end{aligned} \end{aligned}$$

(24)

In the step (a), we use $x = x_+ + x_-$ and $\langle x_-, \check{s}_{\star } \rangle \le 0$. Hence, one can bound $\left\| \mathcal {P}_{\mathcal {F}_{s_\star }^\circ }(x) \right\| _2$ by combining the above bound and the bound on $\mathcal {P}_{\mathcal {V}^\perp }(x_+)$ established in the main text.

Case $s_\star = { \mathbf{S}}_+^n$. For $\mathcal {K}= { \mathbf{S}}_+^{n}$, $\mathcal {P}_{\mathcal {F}_{S_\star }^\circ }(X)$ is

$$\begin{aligned} \mathcal {P}_{\mathcal {F}_{S_\star }^\circ }(X) = X - V(V^\top XV)_+V^\top = X_+ - V(V^\top XV)_+V^\top +X_-. \end{aligned}$$

Since $\mathcal {P}_{\mathcal {V}_{S_\star }^\top }(X+) = X_+ - VV^\top X_+VV^\top$, we know

$$\begin{aligned} \begin{aligned} \left\| \mathcal {P}_{\mathcal {F}_{s_\star }^\circ }(X) - \mathcal {P}_{\mathcal {V}^\perp }(X_+)\right\| _{{\tiny \mathrm{F}}}&= \left\| V(V^\top XV)_+V^\top - VV^\top X_+VV^\top +X_-\right\| _{{\tiny \mathrm{F}}}\\&\overset{(a)}{\le } \left\| (V^\top XV)_+- V^\top X_+V\right\| _{{\tiny \mathrm{F}}} +\left\| X_-\right\| _{{\tiny \mathrm{F}}}\\&\overset{(b)}{\le }2\left\| X_-\right\| _{{\tiny \mathrm{F}}}. \end{aligned} \end{aligned}$$

(25)

In step (a), we use the fact that V has orthonormal columns, and in step (b), we use the fact that $V^\top X_+V$ is still positive semidefinite and projection to the convex set ${ \mathbf{S}}_+^r$ is nonexpansive. Thus, we can bound $\left\| \mathcal {P}_{\mathcal {F}_{s_\star }^\circ }(X)\right\| _{{\tiny \mathrm{F}}}$ using the result for $\left\| \mathcal {P}_{\mathcal {V}^\perp }(X_+)\right\| _{{\tiny \mathrm{F}}}$ in the main text.

Case x is feasible. Finally, note that when x is feasible for ($\mathcal {P}$), then in each of the five cases considered in the paper, the bound (22) and the bound (4) in Theorem 1 coincide.

Proof

Recall Lemma 2 establishes an error bound only for $x\in \mathcal {V}_{s_\star }$. Using the conic decomposition into the face $\mathcal {F}$ and its polar, for any $x\in \mathbf {E}$,

$$\begin{aligned} x = \mathcal {P}_{\mathcal {F}_{s_\star }}(x) + \mathcal {P}_{\mathcal {F}_{s_\star }^{\circ }}(x). \end{aligned}$$

This decomposition immediately gives

$$\begin{aligned} \begin{aligned} \mathbf{dist{}}(x,\mathcal {X}_\star )&\le \left\| \mathcal {P}_{\mathcal {F}_{s_\star }^\circ }(x)\right\| _2 + \mathbf{dist{}}(\mathcal {P}_{\mathcal {F}_{s_\star }}(x), \mathcal {X}_\star ). \end{aligned} \end{aligned}$$

(26)

The second term $\mathbf{dist{}}(\mathcal {P}_{\mathcal {F}_{s_\star }}(x), \mathcal {X}_\star )$ can be bounded using Lemma 2 as $\mathcal {F}_{s_\star } \subset \mathcal {V}_{s_\star }$:

$$\begin{aligned} \mathbf{dist{}}(\mathcal {P}_{\mathcal {F}_{s_\star }}(x), \mathcal {X}_\star )\le \gamma \left\| \mathcal {A}(\mathcal {P}_{\mathcal {F}_{s_\star }}(x))-b\right\| _2. \end{aligned}$$

(27)

Using the decomposition $x = \mathcal {P}_{\mathcal {F}_{s_\star }}(x) + \mathcal {P}_{\mathcal {F}_{s_\star }^{\circ }}(x)$ again for the term $\left\| \mathcal {A}(\mathcal {P}_{\mathcal {F}_{s_\star }}(x))-b\right\| _2$, we reach the bound (22).

Appendix E: Numerical simulation for the bound (8)

Here we numerically verify the correctness of Inequality (8) for feasible x:

$$\begin{aligned} \begin{aligned} \mathbf{dist{}}(x,\mathcal {X}_\star )&\le (1+\gamma \sigma _{\max }(\mathcal {A}))f(\epsilon _{{\tiny opt}}(x),\left\| x\right\| ). \end{aligned} \end{aligned}$$

(28)

The function f can be found in Table 1.

Experiment setup We generated a random instance $\mathcal {A},b,c$ for each of LP, SOCP, and SDP. We solved the corresponding conic problem and obtained the optimal solution $x_\star$ and a dual optimal $s_\star$. We numerically verified that the strict complementarity (by checking Definition 3 for LP and SDP and (DSC) for SOCP) and the uniqueness of the primal (by checking whether $\sigma _{\min }(\mathcal {A}_{\mathcal {V}_{\star }})>0$) both hold for the three cases. We compute $\gamma = \frac{1}{\sigma _{\min }(\mathcal {A}_{\mathcal {V}_{\star }})}$ according to Lemma 2. Next, we randomly perturbed the solution $x_\star$ 70 many times and obtained (possibly infeasible) $x_i', i = 1,\dots , 70$. We then projected $x_i'$ to the feasible set to obtain $x_i$. Finally, we plotted the suboptimality of $x_i$ versus the distance to $x_\star$ (in blue), and the the suboptimality of $x_i$ versus the bound $(1+\gamma \sigma _{\max }(\mathcal {A}))f(\epsilon _{{\tiny opt}}(x_i),\left\| x_i\right\| )$ (in red) in Fig. 2.

From Fig. 2, we observe that the bounds are valid, as they lie uniformly above the true distance, so (28) holds. The bounds for SDP appear to be looser compared to the bounds for LP and SOCP.

Fig. 2

Verification of the inequality (28). The asserted bound in red is $(1+\gamma \sigma _{\max }(\mathcal {A}))f(\epsilon _{{\tiny opt}}(x),\left\| x\right\| )$. The blue points represent suboptimality versus distance to solution (color figure online)